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

1 Introduction

The problem of solving polynomial equations in one variable, i.e., equations of the form

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 $a_{1},\ldots,a_{n}$. 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 $a_{1},\ldots,a_{n}$. 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 $a_{1},\ldots,a_{n}$. For a general polynomial of degree $n\geqslant 2$, 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 $m$th 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 $n\geqslant 5$ 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 $a_{1},\ldots,a_{n}$, 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 [⁠16⁠ A. Chen, Y.-H. He and J. McKay, Erland Samuel Bring’s “Transformation of Algebraic Equations”, arXiv:1711.09253 (2017) ] 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 $a_{1},\ldots,a_{n}$. With this in mind, we define the resolvent degree $\operatorname{rd}(f)$ of a polynomial $f(x)$ in (⁠1⁠) as the smallest positive integer $r$ such that every root of $f(x)$ can be obtained from $a_{1},\ldots,a_{n}$ 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 $\operatorname{rd}(f)$ by $\operatorname{rd}(n)$, as $f(x)$ ranges over all polynomials of degree $n$. The algebraic form of Hilbert’s 13th problem asks for the value of $\operatorname{rd}(n)$.

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 $\operatorname{crd}(n)$, where “c” stands for “continuous”. Specifically, Hilbert asked whether or not $\operatorname{crd}(7)=3$. In this form, Hilbert’s 13th problem was solved by Kolmogorov [⁠37⁠ A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables. Dokl. Akad. Nauk SSSR (N.S.)108, 179–182 (1956) ] and Arnold [⁠1⁠ V. I. Arnol'd, On functions of three variables. Dokl. Akad. Nauk SSSR114, 679–681 (1957) ] in 1957.⁠¹⁠Arnold was a 19 year old undergraduate student in 1957. He later said that all of his numerous subsequent contributions to mathematics were, in one way or another, motivated by Hilbert’s 13th problem; see [⁠2⁠ V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) ]. They showed that, contrary to Hilbert’s expectation, $\operatorname{crd}(n)=1$ 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 $\operatorname{rd}(n)$, not $\operatorname{crd}(n)$. They point to the body of work on $\operatorname{rd}(n)$ going back centuries before Hilbert (see, e.g., [⁠21⁠ J. Dixmier, Histoire du 13e problème de Hilbert. In Analyse diophantienne et géométrie algébrique, Cahiers Sém. Hist. Math. Sér. 2, Univ. Paris VI, Paris, 85–94 (1993) ]) and to Hilbert’s own 20th century writings, where only $\operatorname{rd}(n)$ was considered (see, e.g., [⁠31⁠ D. Hilbert, Über die Gleichung neunten Grades. Math. Ann.97, 243–250 (1927) ]). Arnold himself was a strong proponent of this point of view [⁠13⁠ F. E. Browder (ed.), Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, American Mathematical Society, Providence (1976) , pp. 45–46], [⁠2⁠ V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) ].

Progress on the algebraic form of Hilbert’s 13th problem has been slow. From what I said above, $\operatorname{rd}(n)=1$ when $n\leqslant 5$; this was known before Hilbert and even before Galois. The value of $\operatorname{rd}(n)$ remains open for every $n\geqslant 6$, and the possibility that $\operatorname{rd}(n)=1$ for every $n$ has not been ruled out. The best known upper bounds on $\operatorname{rd}(n)$ are of the form $\operatorname{rd}(n)\leqslant n-\alpha(n)$, where $\alpha(n)$ 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 J. Wolfson, Tschirnhaus transformations after Hilbert. Enseign. Math.66, 489–540 (2020) ], Sutherland [⁠60⁠ A. J. Sutherland, Upper bounds on resolvent degree and its growth rate. arXiv:2107.08139 (2021) ] and Heberle–Sutherland [⁠30⁠ C. Heberle and A. J. Sutherland, Upper bounds on resolvent degree via sylvester’s obliteration algorithm. arXiv:2110.08670 (2021) ].

There is another reading of the 13th problem, to the effect that Hilbert meant to allow global multi-valued continuous functions; see [⁠2⁠ V. I. Arnol'd, From Hilbert’s superposition problem to dynamical systems. Amer. Math. Monthly111, 608–624 (2004) , p. 613]. These behave in many ways like algebraic functions. If we denote the resolvent degree in this sense by $\operatorname{Crd}(n)$, where “C” stands for “global continuous”, then

As far as I am aware, nothing else is known about $\operatorname{Crd}(n)$ or $\operatorname{rd}(n)$ for $n\geqslant 6$.

On the other hand, in recent decades, considerable progress has been made in studying a related but different invariant, the essential dimension.⁠²⁠The term “essential dimension” was coined by Joe Buhler. The term “resolvent degree” was introduced by Richard Brauer in [⁠8⁠ R. Brauer, On the resolvent problem. Ann. Mat. Pura Appl. (4)102, 45–55 (1975) ]. Joe Buhler and I [⁠14⁠ J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] introduced this notion in the late 1990s. In special instances, it came up earlier, e.g., in the work of Kronecker [⁠38⁠ L. Kronecker, Ueber die Gleichungen fünften Grades. J. Reine Angew. Math.59, 306–310 (1861) ], Klein [⁠35⁠ F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree. Revised ed., Dover Publications, New York (1956) ], Chebotarev [⁠15⁠ N. G. Chebotarev, The problem of resolvents and critical manifolds. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]7, 123–146 (1943) ], Procesi [⁠48⁠ C. Procesi, Non-commutative affine rings. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8)8, 237–255 (1967) ]⁠³⁠Procesi asked about the minimal number of independent parameters required to define a generic division algebra of degree $n$. In modern terminology, this number is the essential dimension of the projective linear group $\mathrm{PGL}_{n}$. and Kawamata [⁠34⁠ Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math.363, 1–46 (1985) ]⁠⁴⁠Kawamata defined an invariant $\operatorname{Var}(f)$ of an algebraic fiber space $f\colon X\to S$, which he informally described as “the number of moduli of fibers of $f$ in the sense of birational geometry”. In modern terminology, $\operatorname{Var}(f)$ is the essential dimension of $f$.. Our focus in [⁠14⁠ J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] 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 pre-existing 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 [⁠25⁠ B. Farb and J. Wolfson, Resolvent degree, Hilbert’s 13th problem and geometry. Enseign. Math.65, 303–376 (2019) ] but focusing on connected, rather than finite groups. The quantity $\operatorname{rd}(n)$ defined above is recovered in this setting as $\operatorname{rd}(\mathrm{S}_{n})$. For more comprehensive surveys of essential dimension and resolvent degree, see [⁠41⁠ A. S. Merkurjev, Essential dimension: a survey. Transform. Groups18, 415–481 (2013) , ⁠51⁠ Z. Reichstein, Essential dimension. In Proceedings of the International Congress of Mathematicians, Vol. II (Hyderabad, India, 2010), Hindustan Book Agency, New Delhi, 162–188 (2011) ] and [⁠25⁠ B. Farb and J. Wolfson, Resolvent degree, Hilbert’s 13th problem and geometry. Enseign. Math.65, 303–376 (2019) ], respectively.

2 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 $k\subset\nobreak K_{0}\subset K$ if there exists a finite-dimensional $K_{0}$-algebra $L_{0}$ such that $L=L_{0}\otimes_{K_{0}}K$. Equivalently, recall that, for any choice of an $K$-vector space basis $e_{1},\ldots,e_{n}$ of $L$, one can encode multiplication in $L$ into the $n^{3}$ structure constants $c_{ij}^{h}\in K$ given by $e_{i}e_{j}=\sum_{h=1}^{n}c_{ij}^{h}e_{h}$. Then $L$ descends to $K_{0}\subset K$ if and only if there exists a basis $e_{1},\ldots,e_{n}$ such that all of the structure constants $e_{ij}^{h}$ with respect to this basis lie in $K_{0}$. The essential dimension $\operatorname{ed}_{k}(L/K)$ is defined as the minimal value of the transcendence degree $\operatorname{trdeg}_{k}(K_{0})$, where $L$ descends to $K_{0}$. If the reference to the base field $k$ is clear from the context, we will write $\operatorname{ed}$ in place of $\operatorname{ed}_{k}$.

If $f(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n}$ is a polynomial over $K$, for some $a_{1},\ldots,a_{n}$, as in (⁠1⁠), we define $\operatorname{ed}_{k}(f)$ as $\operatorname{ed}_{k}(L/K)$, where $L=K[x]/(f(x))$. 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 $\overline{y}\in L$ which generates $L$ as an $K$-algebra and such that the minimal polynomial $g(y)=y^{n}+b_{1}y^{n-1}+\cdots+b_{n}$ of $\overline{y}$ 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 $c_{0},c_{1},\ldots,c_{n-1}\in K$. Here $\overline{x}\in L$ 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 $\leqslant n-1$. In his 1683 paper [⁠62⁠ E. W. Tschirnhaus, A method for removing all intermediate terms from a given equation. ACM SIGSAM Bulletin, 37, 1–3 (2003) ], 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 $\geqslant 5$ 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 $\operatorname{ed}(f)$ taken over all field extensions $K/k$ and all separable polynomials $f(x)\in K[x]$ of degree $n$ by $\operatorname{ed}_{k}(n)$. Kronecker [⁠38⁠ L. Kronecker, Ueber die Gleichungen fünften Grades. J. Reine Angew. Math.59, 306–310 (1861) ] and Klein [⁠35⁠ F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree. Revised ed., Dover Publications, New York (1956) ] showed that

This classical result is strengthened in [⁠14⁠ J. Buhler and Z. Reichstein, On the essential dimension of a finite group. Compositio Math.106, 159–179 (1997) ] as follows.

I recently learned that a variant of the inequality $\operatorname{ed}_{\mathbb{C}}(n)\geqslant\bigl\lfloor\frac{n}{2}\bigr\rfloor$ was known to Chebotarev [⁠15⁠ N. G. Chebotarev, The problem of resolvents and critical manifolds. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]7, 123–146 (1943) ] as far back as 1943.

The problem of finding the exact value of $\operatorname{ed}(n)$ may be viewed as being analogous to Hilbert’s 13th problem with $\operatorname{rd}(n)$, $\operatorname{crd}(n)$ or $\operatorname{Crd}(n)$ replaced by $\operatorname{ed}(n)$. Since Hilbert specifically asked about $\operatorname{rd}(7)$, 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 $\operatorname{ed}_{k}(n+2)\geqslant\operatorname{ed}_{k}(n)+1$ from Theorem ⁠1⁠, we can slightly strengthen (⁠6⁠) in characteristic $0$ as follows:

Beyond (⁠7⁠), nothing is known about $\operatorname{ed}_{\mathbb{C}}(n)$ for any $n\geqslant 8$. I will explain where I think the difficulty lies in Section ⁠5⁠.

Analogous questions can be asked about polynomials that are not separable, assuming $\operatorname{char}(k)=p>0$. In this setting, the role of the degree is played by the “generalized degree” $(n,\mathbf{e})$. Here $n=[S:K]$, where $S$ is the separable closure of $K$ in $L=K[x]/(f(x))$ and $\mathbf{e}=(e_{1},\dots,e_{r})$ is the so-called type of the purely inseparable algebra $L/S$ defined as follows. Given $x\in L$, let us define the exponent $\exp(x,S)$ to be the smallest integer $e$ such that $x^{p^{e}}\in S$. Then $e_{1}$ is the largest value of $\exp(x,S)$ as $x$ ranges over $L$. Choose an $x_{1}\in L$ of exponent $e_{1}$, and define $e_{2}$ as the largest value of $\exp(x,S[x_{1}])$. Now choose $x_{2}\in L$ of exponent $e_{2}$, and define $e_{3}$ as the largest value of $\exp(x,S[x_{1},x_{2}])$, etc. We stop when $S[x_{1},\ldots,x_{r}]=L$. By a theorem of Pickert, the resulting integer sequence $e_{1},\ldots,e_{r}$ satisfies $e_{1}\geqslant\cdots\geqslant e_{r}\geqslant 1$ and does not depend on the choice of the elements $x_{1},\ldots,x_{r}$. One can now define $\operatorname{ed}_{k}(n,\mathbf{e})$ by analogy with $\operatorname{ed}_{k}(n)$: $\operatorname{ed}_{k}(n,\mathbf{e})$ is the maximal value of $\operatorname{ed}_{k}(f)$, as $K$ ranges over all field extension of $k$ and $f(x)\in K[x]$ ranges over all polynomials of generalized degree $(n,\mathbf{e})$. Surprisingly, the case where $\mathbf{e}\neq\emptyset$ (i.e., the polynomials $f(x)$ in question are not separable) turns out to be easier. We refer the reader to [⁠53⁠ Z. Reichstein and A. K. Shukla, Essential dimension of inseparable field extensions. Algebra Number Theory13, 513–530 (2019) ], where an exact formula for $\operatorname{ed}(n,\mathbf{e})$ is obtained.

3 Essential dimension of a functor

Following Merkurjev [⁠6⁠ G. Berhuy and G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math.8, 279–330 (2003) ], 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 $\mathcal{F}$ be a covariant functor from the category of field extensions $K/k$ to the category of sets. Any object $\alpha\in\mathcal{F}(K)$ in the image of the natural (“base change”) map $\mathcal{F}(K_{0})\to\mathcal{F}(K)$ is said to descend to $K_{0}$. The essential dimension $\operatorname{ed}_{k}(\alpha)$ is defined as the minimal value of $\operatorname{trdeg}_{k}(K_{0})$, where the minimum is taken over all intermediate fields $k\subset K_{0}\subset K$ such that $\alpha$ descends to $K_{0}$.

For example, consider the functor $\mathrm{Ass}_{n}$ of $n$-dimensional associative algebras given by

For $A\in\mathrm{Ass}_{n}(K)$, the new definition of $\operatorname{ed}_{k}(A)$ 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 $c_{ij}^{h}$. In particular, $A$ descends to the subfield $K_{0}=k(c_{ij}^{h})$ of $K$, and consequently, $\operatorname{ed}_{k}(A)\leqslant n^{3}$.

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 $v,w\in K^{n}$. The form $q$ (or equivalently, $b$) is called degenerate if the linear form $b(v,*)$ is identically zero for some $0\neq v\in K^{n}$. 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 $e_{1},\ldots,e_{n}$ of $K^{n}$, $q$ can be written as

for some $a_{1},\ldots,a_{n}$ in $K$. In particular, we have that $q$ descends to $K_{0}=k(a_{1},\ldots,a_{n})$, and thus $\operatorname{ed}_{k}(q)\leqslant n$. Note that $q$ is non-degenerate if and only if $a_{1},\ldots,a_{n}\neq 0$.

Yet another interesting example is provided by the functor of elliptic curves,

For simplicity, assume that $\operatorname{char}(k)\neq 2$ or $3$. Then every elliptic curve $X$ over $K$ is isomorphic to the plane curve cut out by a Weierstrass equation $y^{2}=x^{3}+ax+b$ for some $a,b\in K$. Hence, $X$ descends to $K_{0}=k(a,b)$ and $\operatorname{ed}(X)\leqslant 2$.

Informally, we think of $\mathcal{F}$ as specifying the type of algebraic object under consideration (e.g., algebras or quadratic forms or elliptic curves), $\mathcal{F}(K)$ as the set of objects of this type defined over $K$, and $\operatorname{ed}_{k}(\alpha)$ as the minimal number of parameters required to define $\alpha$. 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 $\mathcal{F}$. That is,

as $K$ varies over all fields containing $k$ and $\alpha$ varies over $\mathcal{F}(K)$. Note that $\operatorname{ed}_{k}(\mathcal{F})$ can be either a non-negative integer or $\infty$. In particular, the arguments above yield

One can show that the last two of these inequalities are, in fact, sharp. The exact value of $\operatorname{ed}(\mathrm{Ass}_{n})$ is unknown for most $n$; however, for large $n$,

Similarly,

where $\mathrm{Lie}_{n}$ and $\operatorname{Comm}_{n}$ 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 [⁠44⁠ Y. A. Neretin, An estimate for the number of parameters defining an n-dimensional algebra. Izv. Akad. Nauk SSSR Ser. Mat.51, 306–318, 447 (1987) ].⁠⁵⁠Note the resemblance of these asymptotic formulas to the classical theorem of Higman and Sims, which assert that the number of finite $p$-groups of order $p^{n}$ (up to isomorphism) is asymptotically $p^{\smash[b]{2n^{3}/27+O(n^{8/3})}}$. This is not an accident; see [⁠45⁠ B. Poonen, The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS)10, 817–836 (2008) ].

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 $\overline{K}$, $T$ becomes equivariantly isomorphic to $G$ acting on itself by left translations. If $T$ has a $K$-point $x$, then $G\to T$ taking $g$ to $g\cdot x$ 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 $\operatorname{char}(k)\neq 2$, then every $C_{2}$-torsor is of the form $T_{a}$, where $T_{a}$ is the subvariety of $\mathbb{A}^{1}$ cut out by the quadratic equation $x^{2}-a=0$ for some $a\in K$. Informally, $T_{a}$ is a pair of points (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, $H^{1}(K,C_{2})$ is in bijective correspondence with $K^{*}/(K^{*})^{2}$ given by $T_{a}\mapsto a\bmod(K^{*})^{2}$, where $K^{*}$ is the multiplicative group of $K$. Note that, in this example, $H^{1}(K,G)$ is, in fact, a group. 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 $H^{1}(\ast,G)$ parametrizes interesting algebraic objects. For example, when $G$ is the orthogonal group $\mathrm{O}_{n}$, $H^{1}(\ast,\mathrm{O}_{n})$ is the functor $\mathrm{Quad}_{n}$ we considered above. When $G$ is the projective linear group $\mathrm{PGL}_{n}$, $H^{1}(K,\mathrm{PGL}_{n})$ 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 $K$.

4 Essential dimension of an algebraic group

The essential dimension of the functor $H^{1}(*,G)$ is abbreviated as $\operatorname{ed}_{k}(G)$. 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 [⁠12⁠ P. Brosnan and R. Sreekantan, Essential dimension of abelian varieties over number fields. C. R. Math. Acad. Sci. Paris346, 417–420 (2008) ]. If $G$ is the symmetric group $\mathrm{S}_{n}$, then

where $\operatorname{ed}_{k}(n)$ is the quantity we defined and studied in Section ⁠2⁠. Indeed, $H^{1}(K,\mathrm{S}_{n})$ is the set of étale algebras $L/K$ of degree $n$. Étale algebras of degree $n$ are precisely the algebras of the form $K[x]/(f(x))$, where $f(x)$ is a separable (but not necessarily irreducible) polynomial of degree $n$ over $K$. Thus (⁠8⁠) is just a restatement of the definition of $\operatorname{ed}_{k}(n)$.

Another interesting example is the general linear group $G=\mathrm{GL}_{n}$. Elements of $H^{1}(K,\mathrm{GL}_{n})$ are the $n$-dimensional vector spaces over $K$. Since there is only one $n$-dimensional $K$-vector space up to $K$-isomorphism, we see that $H^{1}(K,\mathrm{GL}_{n})=\{1\}$. In particular, every object of $H^{1}(K,\mathrm{GL}_{n})$ descends to $k$, and we conclude that $\operatorname{ed}_{k}(\mathrm{GL}_{n})=0$. I will now give a brief summary of three methods for proving lower bounds on $\operatorname{ed}_{k}(G)$ for various linear algebraic groups $G$.

4.1 Cohomological invariants

Let $\mathcal{F}$ be a covariant functor from the category of field extensions $K/k$ to the category of sets, as in the previous section. A cohomological invariant of degree $d$ for $\mathcal{F}$ is a morphism of functors

for some discrete $\operatorname{Gal}(k)$-module $M$. In many interesting examples, $M=\mu_{m}$ is the module of $m$th roots of unity with a natural $\operatorname{Gal}(k)$-action (trivial if $k$ contains a primitive $m$-th root of unity). The following observation is due to J.-P. Serre.

The proof is an immediate consequence of the Serre vanishing theorem. Cohomological invariants of an algebraic group $G$ (or equivalently, of the functor $H^{1}(\ast,G)$) were introduced by Serre and Rost in the early 1990s, and have been extensively studied since then; see [⁠57⁠ J.-P. Serre, Cohomological invariants, Witt invariants, and trace forms. In Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, American Mathematical Society, Providence, 1–100 (2003) ]. These invariants give rise to a number of interesting lower bounds on $\operatorname{ed}_{k}(G)$ for various groups $G$; in particular,

1. $\operatorname{ed}(\mathrm{O}_{n})\geqslant n$,

2. $\operatorname{ed}(\mathrm{SO}_{n})\geqslant n-1$ for every $n\geqslant 3$,

3. $\operatorname{ed}(G_{2})\geqslant 3$,

4. $\operatorname{ed}(F_{4})\geqslant 5$,

5. $\operatorname{ed}(\mathrm{S}_{n})\geqslant\bigl\lfloor\frac{n}{2}\bigr\rfloor$.

Inequalities (1), (2) and (3) turn out to be exact; (4) is best known, and (5) is best known for even $n$; see (⁠7⁠).

4.2 Finite abelian subgroups

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 $d\geqslant r$. In other words, only non-toral finite abelian subgroups $A$ 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 $\operatorname{ed}(G)$ in many cases, such as $\operatorname{ed}(E_{7})\geqslant 7$ and $\operatorname{ed}(E_{8})\geqslant 9$, where $E_{7}$ denotes the split simply connected exceptional group of type $E_{7}$ and similarly for $E_{8}$.

4.3 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 $\alpha\in H^{2}(K,D)$ has an index, $\operatorname{ind}(\alpha)$, defined as follows. If $D\simeq\mathbb{G}_{m}$, then $\alpha$ is a Brauer class over $K$, and $\operatorname{ind}(\alpha)$ denotes the Schur index of $\alpha$, as usual. In general, we consider the character group $X(D)$ whose elements are homomorphisms $x\colon D\to\mathbb{G}_{m}$. Note that $X(D)$ is a finitely generated abelian group and each character $x\in X(D)$ induces a homomorphism

The index of $\alpha\in H^{2}(K,D)$ is defined as the minimal value of

as $\{x_{1},\ldots,x_{r}\}$ ranges over generating sets of $X(D)$. Here each $(x_{i})_{*}(\alpha)$ lies in $H^{1}(K,\mathbb{G}_{m})$, and $\operatorname{ind}(x_{i})_{*}(\alpha)$ denotes its Schur index, as above. We now define $\operatorname{ind}(G,D)$ as the maximal index of $\alpha\in\nobreak\partial(H^{1}(K,\overline{G}))\subset H^{2}(K,D)$, where the maximum is taken over all field extensions $K/k$, as $\alpha$ ranges over the image $H^{1}(K,\overline{G})$ in $H^{2}(K,D)$.

Part (1) is known as Merkurjev’s index formula. The inequality of part (2) is based on Karpenko’s incompressibility theorem. Part (b) first appeared in [⁠9⁠ P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ] in the special case where $D=\mathbb{G}_{m}$ or $\mu_{p^{r}}$ and in [⁠26⁠ M. Florence, On the essential dimension of cyclic p-groups. Invent. Math.171, 175–189 (2008) ] in an even more special case, where $D=\mu_{p}$. It was proved in full generality in [⁠33⁠ N. A. Karpenko and A. S. Merkurjev, Essential dimension of finite p-groups. Invent. Math.172, 491–508 (2008) ].

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 $\mathrm{Spin}_{n}$. The latter turned out to increase exponentially in $n$:

This inequality was first proved in [⁠9⁠ P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ]. The exact value of $\operatorname{ed}(\mathrm{Spin}_{n})$ subsequently got pinned down in [⁠10⁠ P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension, spinor groups, and quadratic forms. Ann. of Math. (2)171, 533–544 (2010) , ⁠18⁠ V. Chernousov and A. Merkurjev, Essential dimension of spinor and Clifford groups. Algebra Number Theory8, 457–472 (2014) ] in characteristic $0$, [⁠28⁠ S. Garibaldi and R. M. Guralnick, Spinors and essential dimension. Compos. Math.153, 535–556 (2017) ] in characteristic $p\neq 2$ and [⁠61⁠ B. Totaro, Essential dimension of the spin groups in characteristic 2. Comment. Math. Helv.94, 1–20 (2019) ] in characteristic $2$. When $n\geqslant 15$, inequality (⁠9⁠) is sharp for $n\not\equiv 0$ modulo $4$, and is off by $2^{\nu_{2}(n)}$ otherwise. Here $2^{\nu_{2}(n)}$ is the largest power of $2$ dividing $n$.

The exponential growth of $\operatorname{ed}(\mathrm{Spin}_{n})$ came as a surprise. Prior to [⁠9⁠ P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension and algebraic stacks. arXiv:math/0701903 (2007) ], the best known lower bounds on $\operatorname{ed}(\mathrm{Spin}_{n})$ were linear (see [⁠19⁠ V. Chernousov and J.-P. Serre, Lower bounds for essential dimensions via orthogonal representations. J. Algebra305, 1055–1070 (2006) , Section 7]), on the order of $\frac{n}{2}$. Moreover, the exact values of $\operatorname{ed}(\mathrm{Spin}_{n})$ for $n\leqslant 14$ obtained by Rost and Garibaldi [⁠27⁠ S. Garibaldi, Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc.200, xii+81 (2009) ] appeared to suggest that these linear bounds should be sharp. The fact that $\operatorname{ed}(\mathrm{Spin}_{n})$ increases exponentially in $n$ has found interesting applications in the theory of quadratic forms. For details, see [⁠10⁠ P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension, spinor groups, and quadratic forms. Ann. of Math. (2)171, 533–544 (2010) , ⁠18⁠ V. Chernousov and A. Merkurjev, Essential dimension of spinor and Clifford groups. Algebra Number Theory8, 457–472 (2014) ].

5 Essential dimension at $p$

Once again, fix a base field $k$, and let $\mathcal{F}$ be a covariant functor from the category of field extensions $K/k$ to the category of sets. The essential dimension $\operatorname{ed}_{k}(\alpha;p)$ of an object $\alpha\in\mathcal{F}(K)$ at a prime $p$ is defined as the minimal value of $\operatorname{ed}_{k}(\alpha^{\prime};p)$, where the minimum ranges over all finite field extensions $K^{\prime}/K$ of degree prime to $p$ and $\alpha^{\prime}$ denotes the image of $\alpha$ under the natural map $\mathcal{F}(K)\to\mathcal{F}(K^{\prime})$. Finally, the essential dimension $\operatorname{ed}_{k}(\mathcal{F};p)$ of $\mathcal{F}$ at $p$ is the maximal value of $\operatorname{ed}_{k}(\alpha)$, as $K$ ranges over all fields containing $k$ and $\alpha$ ranges over $\mathcal{F}(K)$. When $\mathcal{F}=H^{1}(*,G)$ for an algebraic group $G$, we write $\operatorname{ed}_{k}(G;p)$ in place of $\operatorname{ed}_{k}(\mathcal{F};p)$. Once again, if the reference to the base field is clear from the context, we will abbreviate $\operatorname{ed}_{k}$ as $\operatorname{ed}$. By definition, $\operatorname{ed}(\alpha;p)\leqslant\operatorname{ed}(\alpha)$ and $\operatorname{ed}(\mathcal{F};p)\leqslant\operatorname{ed}(\mathcal{F})$.