From Hilbert’s 13th Problem to essential dimension and back

goes back to ancient times. Here by “solving" I mean finding a procedure or a formula which produces a solution x for a given set of coefficients a1, . . . , an. The terms “procedure" and “formula" are ambiguous; to get a well-posed problem, we need to specify what kinds of operations we are allowed to perform to obtain x from a1, . . . , an. In the simplest setting we are only allowed to perform the four arithmetic operations: addition, subtraction, multiplication and division. In other words, we are asking if the polynomial (1) has a root x which is expressible as a rational function of a1, . . . , an. For a general polynomial of degree n > 2, the answer is clearly “no"; this was already known to the ancient Greeks. The focus then shifted to the problem of “solving polynomials in radicals", where one is allowed to use the four arithmetic operations and radicals of any degree. Here the mth radical (or root) of t is a solution to


Introduction
The problem of solving polynomial equations in one variable, i.e., equations of the form f (x) = 0, where f (x) = x n + a 1 x n−1 + ... + a n , goes back to ancient times. Here by "solving" I mean finding a procedure or a formula which produces a solution x for a given set of coefficients a 1 , . . . , a n . The terms "procedure" and "formula" are ambiguous; to get a well-posed problem, we need to specify what kinds of operations we are allowed to perform to obtain x from a 1 , . . . , a n . In the simplest setting we are only allowed to perform the four arithmetic operations: addition, subtraction, multiplication and division. In other words, we are asking if the polynomial (1) has a root x which is expressible as a rational function of a 1 , . . . , a n . For a general polynomial of degree n 2, the answer is clearly "no"; this was already known to the ancient Greeks. The focus then shifted to the problem of "solving polynomials in radicals", where one is allowed to use the four arithmetic operations and radicals of any degree. Here the mth radical (or root) of t is a solution to Mathematicians attempted to solve polynomial equations this way for centuries, but only succeeded for n = 1, 2, 3 and 4. It was shown by Ruffini, Abel and Galois in the early 19th century that a general polynomial of degree n 5 cannot be solved in radicals. This was a ground-breaking discovery. However, the story does not end there.
Suppose we allow one additional operation, namely solving That is, we start with a 1 , . . . , a n , and at each step we are allowed to enlarge this collection by adding one new number, which is the sum, difference, product or quotient of two numbers in our collection, or a solution to (2) or (3) for any t in our collection. In 1786 Bring [9] showed that every polynomial equation of degree 5 can be solved using these operations. Note that the coefficients of (2) and (3) only depend on one parameter t. Thus roots of these equations can be thought of as "algebraic functions" of one variable. By contrast, the coefficients of the general polynomial equation (1) depend on n independent parameters a 1 , . . . , a n . With this in mind, we define the resolvent degree rd( f ) of a polynomial f (x) in (1) as the smallest positive integer r such that every root of f (x) can be obtained from a 1 , . . . , a n in a finite number of steps, assuming that at each step we are allowed to perform the four arithmetic operations and evaluate algebraic functions of r variables. Let us denote the largest possible value of rd( f ) by rd(n), as f (x) ranges over all polynomials of degree n. The algebraic form of Hilbert's 13th problem asks for the value of rd(n).
The actual wording of the 13th problem is a little different: Hilbert asked for the minimal integer r one needs to solve every polynomial equation of degree n, assuming that at each step one is allowed to perform the four arithmetic operations and apply any continuous (rather than algebraic) function in r variables. Let us denote the maximal possible resolvent degree in this setting by crd(n), where "c" stands for "continuous". Specifically, Hilbert asked whether or not crd(7) = 3. In this form Hilbert's 13th problem was solved by Kolmogorov [36] and Arnold [1] in 1957. 1 They showed that, contrary to Hilbert's expectation, crd(n) = 1 for every n. In other words, continuous functions in 1 variable are enough to solve any polynomial equation of any degree. Moreover, any continuous function in n variables can be expressed as a composition of functions of one variable and addition.
In spite of this achievement Wikipedia lists the 13th problem as "unresolved". While this designation is subjective, it reflects the view of many mathematicians that Hilbert's true intention was to ask about rd(n), not crd(n). They point to the body of work on rd(n) going back centuries before Hilbert (see, e.g., [20]) and to Hilbert's own 20th century writings, where only rd(n) was considered (see, e.g., [30]). Arnold himself was a strong proponent of this point of view [14, pp. 45-46], [2].
Progress on the algebraic form of Hilbert's 13th problem has been slow. From what I said above, rd(n) = 1 when n 5; this was known before Hilbert and even before Galois. The value of rd(n) remains open for every n 6, and the possibility that rd(n) = 1 for every n has not been ruled out. The best known upper bounds on rd(n) are of the form where α(n) is an unbounded but very slow growing function of n. The list of people who have proved inequalities of this form includes some of the leading mathematicians of the past two centuries: Hamilton, Sylvester, Klein, Hilbert, Chebotarev, Segre, Brauer. Recently, their methods have been refined and their bounds sharpened by Wolfson [63], Sutherland [59] and Heberle-Sutherland [29].
There is another reading of the 13th problem, to the effect that Hilbert meant to allow global multi-valued continuous functions; see [2, p. 613]. These behave in many ways like algebraic functions. If we denote the resolvent degree in this sense by Crd(n), where "C" stands for "global continuous", then 1 = crd(n) Crd(n) rd(n) n − α(n).
As far as I am aware, nothing else is known about Crd(n) or rd(n) for n 6.
On the other hand, in recent decades considerable progress has been made in studying a related but different invariant, the essential dimension. 2 Joe Buhler and I [15] introduced this notion in the late 1990s. In special instances it came up earlier, e.g., in the work of Kronecker [37], Klein [34], Chebotarev [61], Procesi [47] 3 and Kawamata [33] 4 . Our focus in [15] was on polynomials and field extensions. It later became clear that the notion of essential dimension is of interest in other contexts: quadratic forms, central simple algebras, torsors, moduli stacks, representations of groups and algebras, etc. In each case it poses new questions about the underlying objects and occasionally leads to solutions of preexisting open problems. This paper has two goals. The first is to survey some of the research on essential dimension in Sections 2-7. This survey is not comprehensive; it is only intended to convey the flavor of the subject and sample some of its highlights. My second goal for this paper is to define the notion of resolvent degree of an algebraic group in Section 8, building on the work of Farb and Wolfson [24] but focusing on connected, rather than finite groups. The quantity rd(n) defined above is recovered in this setting as rd(S n ). For more comprehensive surveys of essential dimension and resolvent degree, see [40,50] and [24], respectively.
I am grateful EMS Magazine's editor Vladimir L. Popov for asking me to write a survey on essential dimension and resolvent degree, and for his help with the exposition. I would also like to thank Mikhail Borovoi for stimulating discussions concerning Conjecture 17, and Patrick Brosnan, Danny Ofek, Federico Scavia and Jesse Wolfson for constructive comments.

Essential dimension of a polynomial
Let k be a base field, K be a field containing k and L be a finite-dimensional K-algebra (not necessarily commutative, associative or unital). We say that L descends to an intermediate field k ⊂ K 0 ⊂ K if there exists a finite-dimensional K 0 -algebra L 0 such that L = L 0 ⊗ K 0 K. Equivalently, recall that for any choice of an K-vector space basis e 1 , . . . , e n of L one can encode multiplication in L into the n 3 structure constants c h i j ∈ K given by e i e j = n h=1 c h i j e h . Then L descends to K 0 ⊂ K if and only if there exists a basis e 1 , . . . , e n such that all of the structure constants e h i j with respect to this basis lie in K 0 . The essential dimension ed k (L/K) is defined as the minimal value of the transcendence degree trdeg k (K 0 ), where L descends to K 0 . If the reference to the base field k is clear from the context, we will write ed in place of ed k .
If f (x) = x n + a 1 x n−1 + . . . + a n is a polynomial over K, for some a 1 , . . . , a n , as in (1) Note that if f (x) (or equivalently, L) is separable over K, then L descends to K 0 if and only if there exists an element y ∈ L which generates L as an K-algebra and such that the minimal polynomial g(y) = y n + b 1 y n−1 + . . . + b n of y lies in K 0 [y].
In classical language the passage from f (x) to g(y) is called a Tschirnhaus transformation. Note that for some c 0 , c 1 , . . . , c n−1 ∈ K. Here x ∈ L is x modulo ( f (x)). Tschirnhaus' strategy for solving polynomial equations in radicals by induction on degree was to transform f (x) to a simpler polynomial g(y), find a root of g(y) and then recover a root of f (x) from (4) by solving a polynomial equation of degree n − 1. In his 1683 paper [62], Tschirnhaus successfully implemented this strategy for n = 3 but made a mistake in implementing it for higher n. Tschirnhaus did not know that a general polynomial of degree 5 cannot be solved in radicals or that his method for solving cubic polynomials had been discovered by Cardano a century earlier.
Let us denote the maximal value of ed( f ) taken over all field extensions K/k and all separable polynomials f (x) ∈ K[x] of degree n by ed k (n). Kronecker [37] and Klein [34] showed that ed C (5) = 2.
This classical result is strengthened in [15] as follows.
and ed k (n + 2) ed k (n) + 1 for every n 1. In particular, for every n 5.
I recently learned that a variant of the inequality ed C (n) n/2 was known to Chebotarev [61] as far back as 1943.
The problem of finding the exact value of ed(n) may be viewed as being analogous to Hilbert's 13th Problem with rd(n), crd(n) or Crd(n) replaced by ed(n). Since Hilbert specifically asked about rd(7), the case, where n = 7 is of particular interest.
The proof of Theorem 2 relies on the same general strategy as Klein's proof of (5); I will discuss it further it in Section 6. Combining Theorem 2 with the inequality ed k (n + 2) ed k (n) + 1 from Theorem 1, we can slightly strengthen (6) in characteristic 0 as follows: n + 1 2 ed(n) n − 3 for every n 7.
Beyond (7), nothing is known about ed C (n) for any n 8. I will explain where I think the difficulty lies in Section 5. Analogous questions can be asked about polynomials that are not separable, assuming char(k) = p > 0. In this setting the role of the degree is played by the "generalized degree" (n, e). Here n = [S : K], where S is the separable closure of K in L = K[x]/( f (x)) and e = (e 1 , . . . , e r ) is the so-called type of the purely inseparable algebra L/S defined as follows. Given x ∈ L, let us define the exponent exp(x, S ) to be the smallest integer e such that x p e ∈ S . Then e 1 is the largest value of exp(x, S ) as x ranges over L. Choose an x 1 ∈ L of exponent e 1 and define e 2 as the largest value of exp(x, S [x 1 ]). Now choose x 2 ∈ L of exponent e 2 and define e 3 as the largest value of exp(x, S [x 1 , x 2 ]), etc. We stop when S [x 1 , . . . , x r ] = L. By a theorem of Pickert the resulting integer sequence e 1 , . . . , e r satisfies e 1 . . . e r 1 and does not depend on the choice of the elements x 1 , . . . , x r . One can now define ed k (n, e) by analogy with ed k (n): ed k (n, e) is the maximal value of ed k ( f ), as K ranges over all field extension of k and f (x) ∈ K[x] ranges over all polynomials of generalized degree (n, e). Surprisingly, the case where e ∅ (i.e., the polynomials f (x) in question are not separable), turns out to be easier. We refer the reader to [52], where an exact formula for ed(n, e) is obtained.
3 Essential dimension of a functor Following Merkurjev [6], we will now define essential dimension for a broader class of objects, beyond polynomials or finite-dimensional algebras. Let k be a base field, which we assume to be fixed throughout, and F be a covariant functor from the category of field extensions K/k to the category of sets. Any object α ∈ F (K) in the image of the natural ("base change") map F (K 0 ) → F (K) is said to descend to K 0 . The essential dimension ed k (α) is defined as the minimal value of trdeg k (K 0 ), where the minimum is taken over all intermediate fields k ⊂ K 0 ⊂ K such that α descends to K 0 .
For example, consider the functor Ass n of n-dimensional associative algebras given by For A ∈ Ass n (K), the new definition of ed k (A) is the same as the definition in the previous section. Recall that after choosing a K-basis for A, we can describe A completely in terms of the n 3 structure constants c h i j . In particular, A descends to the subfield K 0 = k(c h i j ) of K and consequently, ed k (A) n 3 . Another interesting example is the functor of non-degenerate n-dimensional quadratic forms, For simplicity, let us assume that the base field k is of characteristic different from 2. Under this assumption a quadratic form q on K n is the same thing as a symmetric bilinear form b. One passes back and forth between q and b using the formulas for any v, w ∈ K n . The form q (or equivalently, b) is called degenerate, if the linear form b(v, * ) is identically zero for some 0 v ∈ K n . A a variant of the Gram-Schmidt process shows that there exists an orthogonal basis of K n with respect to b. In other words, in some basis e 1 , . . . , e n of K n , q can be written as for some a 1 , . . . , a n in K. In particular, q descends to K 0 = k(a 1 , . . . , a n ) and thus ed k (q) n. Note that q is non-degenerate if and only if a 1 , . . . , a n 0. Yet another interesting example is provided by the functor of elliptic curves, For simplicity assume that char(k) 2 or 3. Then every elliptic curve X over K is isomorphic to the plane curve cut out by a Weierstrass equation Hence, X descends to K 0 = k(a, b) and ed(X) 2.
Informally, we think of F as specifying the type of algebraic object under consideration (e.g., algebras or quadratic forms or elliptic curves), F (K) as the set of objects of this type defined over K, and ed k (α) as the minimal number of parameters required to define α. In most cases essential dimension varies from object to object, and it is natural to consider what happens under a "worst case scenario", i.e., how many parameters are needed to define the most general object of a given type. This number is called the essential dimension of the functor F . That is, as K varies over all fields containing k and α varies over F (K). Note that ed k (F ) can be either a non-negative integer or ∞. In particular, the arguments above yield ed(Ass n ) n 3 , ed(Quad n ) n and ed(Ell) 2.
One can show that the last two of these inequalities are, in fact, sharp. The exact value of ed(Ass n ) is unknown for most n; however, for large n, Similarly ed(Lie n ) = 2n 3 /27 + O(n 8/3 ) and ed(Comm n ) = 2n 3 /27 + O(n 8/3 ), where Lie n and Comm n are the functors of n-dimensional Lie algebras and commutative algebras, respectively. These formulas are deduced from the formulas for the dimensions of the varieties of structure constants for n-dimensional associative, Lie and commutative algebras due to Neretin [43]. 5 This brings us to the functor H 1 ( * , G), where G is an algebraic group defined over k. The essential dimension of this functor is a numerical invariant of G. This invariant has been extensively studied; it will be our main focus in the next section. The functor H 1 ( * , G) associates to a field K/k the set H 1 (K, G) of isomorphism classes of G-torsors T over K. Recall that a G-torsor over T over K is an algebraic variety with a G-action defined over K, such that over the algebraic closure K, T becomes equivariantly isomorphic to G acting on itself by left translations. If T has a K-point x, then G → T taking g to g · x is, in fact, an isomorphism over K. In this case the torsor T is called "trivial" or "split". The interesting (non-trivial) torsors over K have no K-points. For example, if G = C 2 is a cyclic group of order 2 and char(k) 2, then every C 2 -torsor is of the form T a , where T a is the subvariety of A 1 cut out by the quadratic equation x 2 − a = 0, for some a ∈ K. Informally, T a is a pair of point (roots of this equation) permuted by C 2 ; it is split if and only if these points are defined over K (i.e., a is a complete square in K). In fact, This is the case whenever G is abelian. For a non-abelian algebraic group G, H 1 (K, G) carries no natural group structure, it is only a set with a marked element (the trivial torsor).
For many linear algebraic groups G the functor H 1 ( * , G) parametrizes interesting algebraic objects. For example, when G is the orthogonal group O n , H 1 ( * , O n ) is the functor Quad n we considered above. When G is the projective linear group PGL n , H 1 (K, PGL n ) is the set of isomorphism classes of central simple algebras of degree n over K. When G is the exceptional group of type G 2 , H 1 (K, G 2 ) is the set of isomorphism classes of octonion algebras over K.

Essential dimension of an algebraic group
The essential dimension of the functor H 1 ( * , G) is abbreviated as ed k (G). Here G is an algebraic group defined over k. This number is always finite if G is linear but may be infinite if G is an abelian variety [13]. If G is the symmetric group S n , then ed k (S n ) = ed k (n) (8) where ed k (n) is the quantity we defined and studied in Section 2. Indeed, H 1 (K, S n ) is the set of étale algebras L/K of degree n. Étale algebras of degree n are precisely the algebras of the form is a separable (but not necessarily irreducible) polynomial of degree n over K.
Thus (8) is just a restatement of the definition of ed k (n). Another interesting example is the general linear group G = GL n . Elements of H 1 (K, GL n ) are the n-dimensional vector spaces over K. Since there is only one n-dimensional Kvector space up to K-isomorphism, we see that H 1 (K, GL n ) = {1}. In particular, every object of H 1 (K, GL n ) descends to k, and we conclude that ed k (GL n ) = 0. I will now give a brief summary of three methods for proving lower bounds on ed k (G) for various linear algebraic groups G.

Cohomological invariants
Let F be a covariant functor from the category of field extensions K/k to the category of sets, as in the previous section.
for some discrete Gal(k)-module M. In many interesting examples, M = µ m is the module of mth roots of unity with a natural Gal(k)-action (trivial if k contains a primitive m-th root of unity). The following observation is due to J.-P. Serre.
Theorem 3. Assume that the base field k is algebraically closed. If F has a non-trivial cohomological invariant F → H d ( * , M), then ed k (F ) d.
The proof is an immediate consequence of the Serre Vanishing theorem. Cohomological invariants of an algebraic group G (or equivalently, of the functor H 1 ( * , G)) were introduced by Serre and Rost in the early 1990s, and have been extensively studied since then; see [56]. These invariants give rise to a number of interesting lower bounds on ed k (G) for various groups G; in particular, n/2 . Inequalities (i), (ii) and (iii) turn out to be exact. (iv) is best known, and (v) is best known for even n; see (7).

Finite abelian subgroups
Theorem 4. Let G be a reductive group over k and A be a finite abelian subgroup of G of rank r.
(b) ( [28]) Assume char(k) does not divide |A|. If G is connected, and the dimension of the maximal torus of C G (A) is d, then ed(G) r − d.
Note that both parts are vacuous if A lies in a maximal torus T of G. Indeed, in this case the centralizer C G (A) contains T , so d r. In other words, only non-toral finite abelian sugroups A of linear algebraic groups are of interest here. These have been much studied and catalogued, starting with the work of Borel in the 1950s. Theorem 4 yields the best known lower bound on ed(G) in many cases, such as ed(E 7 ) 7 and ed(E 8 ) 9, where E 7 denotes the split simply connected exceptional group of type E 7 and similarly for E 8 .

The Brauer class bound
Consider a linear algebraic group G defined over our base field k. Suppose G fits into a central exact sequence of algebraic groups (again, defined over k) where D is diagonalizable over k. For every field extension K/k, this sequence gives rise to the exact sequence of pointed sets Every element α ∈ H 2 (K, D) has an index, ind(α), defined as follows. If D G m , then α is a Brauer class over K, and ind(α) denotes the Schur index of α, as usual. In general, we consider the character group X(D) whose elements are homomorphisms x : D → G m . Note that X(D) is a finitely generated abelian group and each character x ∈ X(D), induces a homomorphism Part (a) is known as Merkurjev's Index Formula. The inequality of part (b) is based on Karpenko's Incompressibility Theorem. Part (b) first appeared in [10] in the special case where D = G m or µ p r and in [25] in an even more special case, where D = µ p . It was proved in full generality in [32].
Theorem 5 is responsible for some of the strongest results in this theory, including the exact formulas for the essential dimension of a finite p-group (Theorem 6 below), the essential p-dimension of an algebraic torus, and the essential dimension of spinor groups Spin n . The latter turned out to increase exponentially in n: This inequality was first proved in [10]. The exact value of ed(Spin n ) subsequently got pinned down in [11] and [17] in characteristic 0, [27] in characteristic p 2 and [60] in characteristic 2. When n 15, the inequality (10) is sharp for n 0 modulo 4, and is off by a 2 ν 2 (n) otherwise. Here 2 ν 2 (n) is the largest power of 2 dividing n.
The exponential growth of ed(Spin n ) came as a surprise. Prior to [10] the best known lower bounds on ed(Spin n ) were linear (see [18,Section 7]), on the order of n/2. Moreover, the exact values of ed(Spin n ) for n 14 obtained by Rost and Garibaldi [26] appeared to suggest that these linear bounds should be sharp. The fact that ed(Spin n ) increases exponentially in n has found interesting applications in the theory of quadratic forms. For details, see [11] and [17].

Essential dimension at p
Once again, fix a base field k and let F be a covariant functor from the category of field extensions K/k to the category of sets. The essential dimension ed k (α; p) of an object α ∈ F (K) at a prime p is defined as the minimal value of ed k (α ; p), where the minimum ranges over all finite field extensions K /K of degree prime to p and α denotes the image of α under the natural map F (K) → F (K ). Finally, the essential dimension ed k (F ; p) of F at p is the maximal value of ed k (α), as K ranges over all fields containing k and α ranges over F (K). When F = H 1 ( * , G) for an algebraic group G, we write ed k (G; p) in place of ed k (F ; p). Once again, if the reference to the base field is clear from the context, we will abbreviate ed k as ed. By definition, ed(α; p) ed(α) and ed(F ; p) ed(F ).
The reason to consider ed(F ; p) in place of ed(F ), is that the former is often more accessible. In fact, most of the methods we have for proving a lower bound on ed k (α) (respectively, ed k (F )) turn out to produce a lower bound on ed k (α; p) (respectively, ed k (F ; p)) for some prime p. For example, the lower bound in Theorem 5(b) is really ed k (G; p) ind(G, D)− dim(G). In Theorem 4, one can usually choose A to be a pgroup, in which case the conclusion can be strengthened to ed(G; p) r in part (a) and ed(G; p) r − d in part (b). In Theorem 3, if M is p-torsion (which can often be arranged), then ed(G; p) d.
This is a special case of a general meta-mathematical phenomenon: many problems concerning algebraic objects (such as finite-dimensional algebras or polynomials or algebraic varieties) over fields K can be subdivided into two types. In Type 1 problems we are allowed to pass from K to a finite extension K /K of degree prime to p, for one prime p, where as in Type 2 problems this is not allowed. For example, given algebraic variety X defined over K, deciding whether or not X has a 0-cycle of degree 1 is a Type 1 problem (it is equivalent to showing that there is a 0-cycle of degree prime to p, for every prime p), where as deciding whether or not X has a K-point is a Type 2 problem. As I observed in [50,Section 5], most of the technical tools we have are tailor-made for Type 1 problems, where as many long-standing open questions across several areas of algebra and algebraic geometry are of Type 2.
In the context of essential dimension the problem of computing ed(G; p) for a given algebraic group G and a given prime p is of Type 1, where as the problem of computing ed(G) is of Type 2. For simplicity let us assume that G is a finite group. In this case ed k (G; p) = ed k (G p ; p), where G p is the Sylow p-subgroup of G. In other words, the problem of computing ed k (G; p) reduces to the case, where G is a p-group. In this case we we have the following remarkable theorem of Karpenko and Merkurjev [31].
Theorem 6. Let p be a prime and k be a field containing a primitive pth root of unity. Then for any finite p-group P, where rdim k (P) denotes the minimal dimension of a faithful representation of P defined over k.
Theorem 6 reduces the computation of ed k (G; p) to rdim k (G p ). For a given finite p-group P, one can often (though not always) compute rdim k (P) in closed form using the machinery of character theory; see, e.g., [3,35,41,42].
The situation is quite different when it comes to computing ed k (G) for an arbitrary finite group G. Clearly ed k (G) max p ed k (G; p), where p ranges over the prime integers. In those cases, where ed k (G) is strictly larger than max p ed k (G; p), the exact value of ed k (G) is usually difficult to establish. The only approach that has been successful to date relies on classification results in algebraic geometry, which are currently only available in low dimensions. I will return to this topic in the next section.
To illustrate the distinction between Type 1 and Type 2 problems, consider the symmetric group G = S n . For simplicity assume that k = C is the field of complex numbers. Here the Type 1 problem is solved completely: ed C (S n ; p) = n/p for every prime p. Thus max p ed C (S n ; p) = n/2 , and (7) tells us that ed C (S n ) > max p ed C (G; p), for every odd n 7. The remaining Type 2 problem is to bridge the gap between n/2] and the true value of ed C (S n ). This problem has only been solved for n 7; see Theorems 1, 2 and (8).
Note that the algebraic form of Hilbert's 13th problem is also of Type 2 in the sense that rd( f ; p) 1 (11) for any prime p, every field K and every separable polynomial f (x) ∈ K[x]. 6 Indeed, denote the Galois group of f (x) by G. Then after passing from K to a finite extension K /K whose degree [K : K] = [G : G p ] is prime to p, we may replace G by its p-Sylow subgroup G p . Since every p-group is solvable, this means that f (x) becomes solvable in radicals over K and hence, its resolvent degree becomes 1, as desired.
The inequality (11) accounts, at least in part, for the difficulty of showing that rd(n) 2 for any n. The methods used to prove lower bounds on the essential dimension of algebraic groups in Section 4 and anything resembling these methods, cannot possibly work here. Otherwise we would also be able to prove that rd( f ; p) 2 for some prime p, contradicting (11).
A similar situation arises in computing the essential dimension of a finite p-group G over a field k of characteristic p. Superficially this problem looks very different from Hilbert's 13th Problem (where one usually works over k = C); the common feature is that both are Type 2 problems. Indeed, it is shown in [53] that ed k (G; p) = 1 for every non-trivial p-group G. Using the method described in the next section, one can often show that ed k (G) 2, but we are not able to prove that ed k (G) > 2 for any p-group G and any field k of characteristic p. On the other hand, Ledet [38] conjectured that ed k (C p n ) = n (12) for any prime p and any infinite field k of characteristic p.
Here C p n denotes the cyclic group of order p n . Ledet showed that ed k (C p n ) n for every n 1 and that equality holds when n 2.
My general feeling is that Type 2 problems arising in different contexts are linked in some way, and that solving one of them (e.g, proving Ledet's Conjecture) can shed light on the others (e.g., Hilbert's 13th Problem). The only bit of evidence I have in this direction is the following theorem from [12] linking a priori unrelated Type 2 problems in characteristic p and in characteristic 0.
Theorem 7. Let p be a prime and G be a finite group satisfying the following conditions: (i) G does not have a non-trivial normal p-subgroup, and (ii) G has an element of order p n . If Ledet's Conjecture (12) holds, then ed C (G) n.
The following family of examples is particularly striking. Let p be a prime and n a positive integer. Choose a positive integer m such that q = mp n + 1 is a prime. Note that by Dirichlet's theorem on primes in arithmetic progressions, there are infinitely many such m. Let C q be a cyclic group of order q. Then Aut(C q ) is cyclic of order mp n ; let C p n ⊆ Aut(C q ) denote the unique subgroup of order p n . Applying Theorem 7 to G = C q C p n , we obtain the following. Corollary 8. If Ledet's Conjecture (12) holds, then Note that since the Sylow subgroups of C q C p n are all cyclic, ed C (C q C p n ; l) 1 for every prime l, so the inequality of Corollary 8 is a Type 2 result. An unconditional proof of this inequality or even of the weaker inequality ed C (C q C p n ) > 3 is currently out of reach for any specific choice of q and p n .

Essential dimension and the Jordan property
An alternative (equivalent) definition of essential dimension of a finite group G is as follows. An action of G on an algebraic variety X is said to be linearizable if there exists a G-equivariant dominant rational map V X for some linear representation G → GL(V). Then ed k (G) is the minimal value of dim(X), as X ranges over all linearizable varieties with a faithful G-action defined over k. In particular, ed k (G) rdim k (G), (13) where rdim k (G) is the minimal dimension of a faithful linear representation of G over k, as in Theorem 6. This geometric interpretation of ed k (G) can sometimes be used to prove lower bounds on ed(G) by narrowing the possibilities for X and ruling them out one by one using Theorem 4(a). For the remainder of this section I will assume that G is a finite group and the base field k is the field of complex numbers and will write ed in place of ed C .
Suppose ed(G) = 0. Then X is a single point, and only the trivial group can act faithfully on a point. Thus ed(G) = 0 if and only if G is the trivial group. Now suppose ed(G) = 1. Then X is a curve with a dominant map V X. By Lüroth's theorem, X is birationally isomorphic to P 1 and thus G is a subgroup of PGL 2 . Finite subgroups of PGL 2 were classified by Klein [34]. Here is a complete list: cyclic groups C n and dihedral groups D n for every n, A 4 , S 4 and A 5 . Theorem 4(a) rules out the groups on this list which contain A = C 2 × C 2 . We thus obtain the following.
Theorem 9 (Theorem 6.2 in [15]). Let G be a finite group. Then ed(G) = 1 if and only if G is either cyclic or odd dihedral.
To classify groups of essential dimension d (or more realistically, show that ed(G) > d for a particular finite group G) in a similar manner, we need a classification of finite subgroups of Bir(X), extending Klein's classification of finite subgroups in Bir(P 1 ). Here X ranges over the unirational complex varieties of dimension d, and Bir(X) denotes the groups of birational automorphisms of X. In dimension 2 every unirational variety is rational, so we are talking about classifying finite subgroups of the Cremona group Bir(P 2 ). Such a classification exists, though it is rather complicated; see [21]. Serre used this approach to show that ed(A 6 ) = 3 [58, Theorem 3.6]. Again, this is a Type 2 phemenomenon, since max p ed(A 6 ; p) = 2. Duncan [23] subsequently extended Serre's argument to a full classification of finite groups of essential dimension 2.
In dimension 3 there is the additional complication that unirational complex varieties do not need to be rational. Here only a partial analogue of Klein's classification exists, namely the classification of rationally connected 3-folds with the action of a finite simple group G, due to Prokhorov [48]. Duncan used this classification to prove Theorem 2. More specifically, he showed that ed(S 7 ) = ed(A 7 ) = 4; see (8). Subsequently Beauville [4] showed that the only finite simple groups of essential dimension 3 are A 6 and possibly PSL 2 (F 11 ). 7 In dimension d 4 even a partial analogue of Klein's classification of finite subgroups of Bir(P 1 ) is out of reach. However, a recent break-through in Mori Theory gives us a new inisight into the asymptotic behavior of ed(G n ) for certain infinite sequences G 1 , G 2 , . . . of finite groups. Recall that an abstract group Γ is called Jordan if there exists an integer j (called a Jordan constant of Γ) such that every finite subgroup G ⊂ Γ has a normal abelian subgroup A of index [G : A] j. This definition, due to Popov [45], was motivated by the classical theorem of Camille Jordan which asserts that GL n (C) is Jordan, and by a theorem of Serre [57] which asserts that the Cremona group Bir(P 2 ) is also Jordan. The following result, due to Prokhorov, Shramov and Birkar 8 , is a far-reaching generalization of Serre's theorem.
I will say that a collection of abstract groups is uniformly Jordan if they are all Jordan with the same constant.
Then the groups Bir(X) are uniformly Jordan, as X ranges over d-dimensional rationally connected complex varieties.
Unirational varieties are rationally connected. The converse is not known, though it is generally believed to be false. Rationally connected varieties naturally arise in the context of Mori theory, and we are forced to consider them even if we are only really interested in unirational varieties. Note that Theorem 10 does not become any easier to prove if one requires X to be unirational. In fact, prior to Birkar [7] (respectively, prior to Prokhorov-Shramov [48]), it was an open question, due to Serre [57, Section 6], whether for each d 4 (respectively, for d = 3) there exists even a single finite group which does not embed into Bir(P d ). 9 Now observe that while every finite group is obviously Jordan, being uniformly Jordan is a strong condition on a sequence of finite groups Suppose the sequence (14) is chosen so that no infinite subsequence is uniformly Jordan. Then we claim that Indeed, if ed(G n ) = d, then there exists a d-dimensional linearizable variety X with a faithful G n -action. In particular, G n is contained in Bir(X). Since X is linearizable, it is unirational and hence, rationally connected. On the other hand, since no infinite subsequence of (14) is uniformly Jordan, Theorem 10 tells us that there are at most finitely many groups G n with ed(G n ) = d, and (15) follows. Here is an interesting family of examples.
Theorem 11. For each positive integer n, let C n be a cyclic group of order n and H n be a subgroup of Aut(C n ).
Note that this method does not give us any information about ed(C n H n ) for any particular choice of n and of H n ⊂ Aut(C n ). For example, while Theorem 11 tells us that ed C (C p Aut(C p )) > 10 for all but finitely many primes p, it does not allow us to exhibit a specific prime for which this inequality holds. The reason is that when d > 3, a specific Jordan constant for the family of groups Bir(X) in Theorem 10 is out of reach. In particular, an unconditional proof of Corollary 8 along these lines does not appear feasible. Nevertheless, Theorem 11 represents a big step forward: previously it was not even known that ed C ((C p ) Aut(C p )) > 3 for any prime p.
A classification of the subgroups of Bir(X), as X ranges over the unirational varieties of dimension d is a rather blunt instrument. It would be preferable to find some topological or algebro-geometric obstruction to the existence of a linearization map V X, which can be read off from the G-variety X without enumerating all the possibilities for X. Unfortunately, all known obstructions of this sort are of Type 1: they do not distinguish between dominant rational maps V X and correspondences V X of degree prime to p, for a suitable prime p and thus cannot help us if ed(G) > max p ed(G; p).
Another draw-back of this method is that, as we mentioned in the previous section, beyond dimension 1 10 , none of the classification theorems we need are available in prime characteristic.
7 Essential dimension of a representation 7.1 Representations of finite groups in characteristic 0 Let G be a finite group of exponent e, k be a field of characteristic 0, K/k be a field extension, ρ : G → GL n (K) be a representation of G, and χ : G → K be the character of ρ. Can we realize ρ over k? In other words, is there a representation ρ : G → GL(k) such that ρ and ρ are equivalent over K? A celebrated theorem of Richard Brauer asserts that the answer is "yes" as long as k contains a primitive root of unity of degree e. If it doesn't, there is a classical way to quantify how far ρ is from being definable over k via the Schur index, at least in the case where ρ is absolutely irreducible and the character value χ(g) lies in k for every g ∈ G. The Schur index of ρ is defined as the index of the envelope Env k (ρ) := Span k {ρ(g) | g ∈ G} ⊂ Mat n (K) which, under our assumptions on ρ, is a central simple algebra of degree n over k. The Schur index of ρ is equal to the minimal degree [l : k] of a field extension l/k such that ρ can be realized over l.
The essential dimension ed k (ρ) gives us a different way to quantify how far ρ is from being definable over k. Here we do not need to assume that ρ is irreducible or that its character values lie in k. We simply think of ρ as an object of the functor The naive upper bound on ed(ρ) is rn 2 , where n is the dimension of ρ and r is the minimal number of generators of G. Indeed, if G is be generated by r elements g 1 , . . . , g r and ρ(g h ) is the n × n matrix (a h i j ), then ρ descends to the field K 0 = k a h i j | i, j = 1, . . . , n; h = 1, . . . , r of transcendence degree at most rn 2 over k. It is shown in [31] that, in fact, ed(ρ) n 2 /4 and moreover, ed(Rep G ) |G|/4. We have also proved lower bounds on ed k (ρ) in many cases; we refer the reader to [31] for details. Note that these are quite delicate: by Brauer's theorem ed k (G) = 0 as long as k contains suitable roots of unity.

Representations of finite groups in positive characteristic
Here the situation is entirely different.
Theorem 12 ( [31,5]). Let G be a finite group, k be a field of characteristic p > 0 and G p be the Sylow p-subgroup of G. Then Note that by a theorem of Higman, in characteristic p, G p is cyclic if and only if the group algebra kG if of finite representation type, i.e., if and only if kG (or equivalently, G) has only finitely many indecomposible representations. Since kG is always of finite representation type in characteristic 0, we can restate Theorem 12 as follows.
Corollary 13. Let G be a finite group and k be a field of arbitrary characteristic. Then ed k (Rep G ) < ∞, if kG is of finite representation type, and

Representations of algebras
For simplicity let us assume that the base field k is algebraically closed. A celebrated theorem of Drozd asserts that every finite-dimensional k-algebra A falls into one of three categories: (a) finite representation type, (b) tame, and (c) wild. Informally speaking, A is of tame representation type if for every positive integer n, the n-dimensional indecomposable A-modules occur in (at most) a finite number of oneparameter families. On the other hand, A is of wild representation type if the representation theory of A contains that of the free k-algebra on two generators.
We can define the functor of representations Rep A in the same way as before: it associates to a field K/k, isomorphism classes of finite-dimensional A ⊗ k K-modules. Corollary 13 tells us that when A = kG is a group ring, the essential dimension of the functor Rep A distinguishes between algebras A of finite representation type and algebras of the other two types. It does not distinguish between tame and wild representations types, since ed(Rep A ) = ∞ in both cases. Benson suggested that it may be possible to distinguish between these two types of algebras by considering the rate of growth of r A (n) = ed(Rep A [n]), where Rep A [n](K) is the set of isomorphism classes of K-representations of A of dimension n. This is confirmed by the following theorem of Scavia [55].
Theorem 14. (a) If A is of finite representation type, then r A (n) is bounded from above as n → ∞.
(b) If A is tame, then there exists a constant c > 0 such that cn − 1 r A (n) 2n − 1 for every n 1.
(c) If A is wild, then there exist constants 0 < c 1 < c 2 such that c 1 n 2 − 1 r A (n) c 2 n 2 for every n 1.
This gives us three new invariants of finite-dimensional algebras, a i (A) = lim sup n→∞ r A (n)/n i for i = 0, 1, 2. Informally, a 2 (A) (respecitively, a 1 (A)) quantifies "how wild" (respectively, "how tame") A is. Scavia [55] computed a 1 (A) and a 2 (A) explicitly in combinatorial terms in the case, where A is a quiver algebra. 8 Back to resolvent degree

The level of a field extension
Let k be a base field, K be a field containing k, and L/K be a field extension of finite degree. I will say that L/K is of level d if there exists a finite tower of subfields such that L ⊂ K n and ed k (K i+1 /K i ) d for every i. The level of L/K is the smallest such d; I will denote it by lev k (L/K). Clearly lev k (L/K) ed k (L/K). If K is a field of rational functions on some algebraic variety X defined over k, then it is natural to think of elements of K 1 as algebraic (multi-valued) functions on X in ed k (K 1 /K) variables, and elements of L as compositions of algebraic functions in lev k (L/K) variables.
Example 15. If the field extension L/K is solvable, then we claim that lev k (L/K) 1. Indeed, here we can choose the tower (16) so that each K i+1 is obtained from K i by adjoining a single radical. Then ed k (K i+1 /K i ) 1 for each i, and hence, lev k (L/K) 1, as claimed.

The resolvent degree of a functor
Let F be a functor from the category of field extensions K/k to the category of sets with a marked element. We will denote the marked element in F (K) by 1 and will refer to it as being "split". We will say that a field extension L/K splits an object α ∈ F (K) if α L = 1. Here as usual, α L denotes the image of α under the natural map F (K) → F (L). Let us assume that for every field K/k and every α ∈ F (K), α can be (17) split by a field extension L/K of finite degree. This is a strong condition of F ; in particular, it implies that F (K) = {1} whenever K is algebraically closed.
I will now define the resolvent degrees rd k (α) of α ∈ F (K) and rd k (F ) of the functor F satisfying condition (17) by analogy with the definitions of ed k (α) and ed k (F ) in Section 3. The resolvent degree rd k (α) is the minimal integer d 0 such that α is split by a field extension L/K of level d (or equivalently, of level d). The resolvent degree rd k (F ) is the maximal value of rd k (α), as K ranges over all fields containing k and α ranges over F (K).
The resolvent degrees rd k (α; p) and rd k (F ; p) at a prime p are defined in the same way as ed k (α; p) and ed k (F ; p). Here F is a functor satisfying (17) α ∈ F (K) is an object of F . That is, rd k (α; p) is the minimal value of rd k (α K ), as K ranges over all field extension of K such that [K : K] is finite and prime to p, and rd k (F ; p) is the maximal value of rd k (α; p), where K ranges over all field containing k, and α ranges over F (K). A variant of the argument we used to prove (11) shows that rd k (F ; p) 1 for every base field k, every functor F satisfying (17) and every prime p.

The resolvent degree of an algebraic group
The functor F = H 1 ( * , G) whose objects over K are G-torsors over Spec(K) satisfies condition (17) for every algebraic group G defined over k. I will write rd k (G) for the resolvent degree of this functor. For simplicity let us assume that k = C for the remainder of this section. I will write rd in place of rd C .
Note that the quantity rd(n) we defined in the Introduction can be recovered in this setting as rd(S n ); cf. (8). Moreover, for a finite group G our definition of rd(G) coincides with the definition given by Farb and Wolfson in [24].
Recall that for a polynomial f (x) ∈ K[x], our definition of rd( f ) was motivated by wanting to express a root of f (x) as a composition of algebraic functions in d variables applied to the coefficients. Equivalently, we wanted to find the smallest d such that the 0-cycle in A 1 K given by f (x) = 0 has an L-point, for some field extension L/K of level d. If G is a linear algebraic group and T → Spec(K) is a G-torsor, then our more general definition of rd(T ) retains this flavor. Indeed, T is an affine variety defined over K, and saying that T is split by L is the same as saying that T has an L-point.
While little is known about rd(n) = rd(S n ), it is natural to ask what rd(G) is for other algebraic groups G. Such questions can be thought of as variants of Hilbert's 13th Problem. Let us now take a closer look at the case, where G is linear and connected. The following folklore conjecture is implicit in the work of Tits.
Conjecture 17. Let G be a connected complex linear algebraic group and K be a field containing C. Then every α ∈ H 1 (K, G) is split by some solvable field extension L/K. Since solvable extensions are of level 1, this conjecture implies that rd(G) 1 for every connected linear algebraic group G. 11 I can prove the following weaker inequality unconditionally [51].
Theorem 18. Let G be a connected complex linear algebraic group. Then rd(G) 7.
Note that if we knew that rd(S n ) d for every n, we would be able to conclude that lev(L/K) d for every field extension L/K of finite degree. This would, in turn, imply that rd(F ) d for every functor F satisfying (17). Setting F = H 1 ( * , G), we obtain rd(G) d for every algebraic group G. In particular, if we were able to show that rd(F ) > 1 for some functor F satisfying (17), we would be able to conclude that rd(S n ) > 1 for some n. This would constitute major progress on Hilbert's 13th problem. I do not see how to reverse this implication though: an upper bound on rd(G) for every connected group G (such as the inequality rd(G) 7 of Theorem 18) does not appear to tell us anything about rd(S n ). However, Conjecture 17 and Theorem 18 make me take more seriously the possibility that rd(S n ) may be identically 1 or at least bounded as n → ∞.