Algebraic theorems with no known algebraic proofs

Question

What are some good examples of algebraic theorems that have no known algebraic proofs?

A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the theorem due to Hopf that the only commutative ones have dimensions 1 or 2 (like $\mathbb{R}$ or $\mathbb{C}$); and the theorem due to Kervaire/Milnor that the only (not necessarily commutative) ones have dimensions 1, 2, 4, or 8 (like $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, or $\mathbb{O}$). At least, the Wikipedia page on suggests there are not known algebraic proofs of these theorems: all known proofs use topology or geometry.

Answer 1

Here is my favorite one (though not so elementary).

Theorem (Grothendieck). Let $X$ be a smooth projective variety over an algebraically closed field $k$. Then, the etale fundamental group $\pi^{\rm et}_1(X)$ is topologically finitely generated.

Note that $k$ might have characteristic $p$. Nevertheless, all known proofs of this fact eventually reduce to the case of a smooth projective curve over $\mathbb{C}$, and use the comparison theorem with the topological fundamental group.

An algebraic proof of the topological finite generation is not known even for the prime-to-$p$ completion of $\pi_1^{\rm et}(X)$. I think we know how to reduce this possibly easier question to the special case of the single (non-projective) variety $X= \mathbb{P}^1_k \setminus \{0,1,\infty\}$. Still, even in this case we need the complex numbers!

(For $\ell$-adic cohomology, there are general algebraic proofs of finite dimensionality. But not for the fundamental group!)

Answer 2

An example comes from the connection of function fields and Riemann surfaces: Let $K$ be an algebraically closed field of characteristic $0$, $t$ a variable, and $L/K(t)$ be a finite Galois extension. Then for each place $p_i$ of $K(t)$ which is ramified in $L$ there is a place $P_i$ of $L$ above $p_i$ whose (cyclic) inertia group is generated by $\sigma_i$ such that the following holds:

• The Galois group of $L/K(t)$ is generated by the $\sigma_i$.
• $\sigma_1\sigma_2\cdots\sigma_r=1$.

As far as I know, the only way to prove this purely algebraic statement is to first reduce it to $K=\mathbb C$, and then reformulate it in terms of a branched covering of compact Riemann surfaces and making use of the fundamental group of a punctured Riemann sphere.

The converse, by the way, holds too, and relies on the same translation.

Added 2024-11-22: David Harbater's beautiful survey article Riemann’s Existence Theorem details the failed attempts to prove either direction algebraically, and also discusses weaker results which were shown algebraically using formal schemes or rigid analytic geometry (like Florian Pop's Half Riemann’s Existence Theorem) over more general fields.

Answer 3

Artin's theorem on positive polynomials, which solves Hilbert's 17th problem in the affirmative, apparently still has no algebraic proof.

Theorem (Artin): If $f \in \mathbb{R}[X_1,\dots,X_n]$ is pointwise nonnegative, then it is a sum of squares in $\mathbb{R}(X_1,\dots,X_n)$.

Artin's proof uses quantifier elimination for real closed fields from model theory, so this arguably qualifies as not purely algebraic. As far as I know this is still the only successful approach to the problem.

Answer 4

There should be many examples in algebraic number theory, as it often interacts nontrivially with analytic number theory. For example, I am pretty sure there is no algebraic proof of Bauer's theorem (but would love to be proved wrong):

Theorem (Bauer, 1903) Let $K \to L$ and $K \to M$ be finite Galois extensions of number fields, and assume that a prime $\mathfrak p \subseteq \mathcal O_K$ is totally split in $L$ if and only if it is totally split in $M$. Then $L \cong M$.

This follows pretty easily from the Chebotarev density theorem (or Kronecker's split version, see Corollary 2.4 of these notes by Keith Conrad).

It's not just the proof of the Chebotarev density theorem that uses analytic number theory, somehow the statement itself is analytic in nature. But its corollaries, such as Bauer's theorem, may be purely algebraic statements. (I would not be surprised if there were other key theorems with these attributes, for instance in geometric group theory.)

Answer 5

The Scott-Wiegold conjecture posits that the free product of three non-trivial cyclic groups can never be normally generated by a single element.

It was eventually proven by Jim Howie, by studying the homotopy theory of the representation variety of $F_3$ in $SU_2$.

Similar uses of homotopy theory to solve purely algebraic problems:

1. The finite group case of the Kervaire conjecture.

2. The non-freeness of the module $A^3/\left(\begin{array}{c}x\\y\\z\end{array}\right)$, where $A=\mathbb{Z}[x,y,z]/\langle 1-x^2-y^2-z^2\rangle$, usually proved via the Hairy Ball theorem.

3. Let $R=\mathbb{Z}[x,y,t]/\langle 1-x^2-y^2\rangle$. Then the elements $$\alpha= x+(1-t^2)a,\\ \beta=yt+(1-t^2)b,$$ can never generate $R$ as an ideal, for any $a,b\in R$. Easily proved by considering the winding numbers (at $t=1$ and $t=-1$) of $(\alpha,\beta)$ about the origin in $\mathbb{R}^2$, as $(x,y)$ goes round the unit circle in $\mathbb{R}^2$.

Answer 6

The fundamental theorem of algebra states that every non-constant single-variable polynomial with coefficients in $\mathbb C$ has at least one complex root; i.e., the field of complex numbers is algebraically closed. Known proofs involve some mathematical analysis, or at least the topological concept of continuity.

Answer 7

The proof (by Hecke I believe) that in a number field the ideal class of the different is always the square of a class.

Answer 8

Dirichlet's theorem on primes in arithmetic progression: if $a$ and $b$ are any relatively prime positive integers, then there are infinitely many prime numbers of the form $a+nb$ for positive integers $n$.

EDIT, given @SamHopkins comment: Dirichlet's theorem, states, equivalently, that a linear polynomial in the UFD $\mathbb{Z}[X]$ represents infinitely many irreducible elements of the UFD $\mathbb{Z}$ if and only if the polynomial is nonconstant and irreducible over $\mathbb{Z}$. One can also include the generalization of Dirichlet's theorem to number fields. Dirichlet's unit theorem is another example.

Answer 9

Okninski proved that $M$ is a monoid and $K$ is a field of characteristic $0$ such that the monoid algebra $KM$ is von Neumann regular, then $M$ is locally finite using analytic methods. No algebraic proof is known. Recall a ring $R$ is von Neumann regular if $\forall a\in R, \exists b\in R$ such that $aba=a$ and a monoid is locally finite if all its finitely generated submonoids are finite.

The proof is quite short. Let $X$ be a finite subset of $M$. We want the submonoid $\langle X\rangle$ is finite. Let $a=\frac{1}{|X|+1}\sum_{x\in X} x\in \mathbb QM\subseteq KM$. Let $b\in KM$ with $(1-a)b(1-a)=1-a$. The subfield $L$ of $K$ generated by the coefficients of $a,b$ is finitely generated and hence embeds in $\mathbb C$. Thus we may view $1-a,a,b\in \mathbb CM$. Now $\mathbb CM$ embeds in the Banach algebra $\ell_1M$. The element $a$ has $\ell_1$-norm $\|a\|_1<1$. Thus in $\ell_1M$ we have that $1-a$ is a unit with inverse $\sum_{n=0}^\infty a^n$. We deduce that $b=(1-a)^{-1}$. But $b$ has finite support whereas $(1-a)^{-1}=\sum_{n=0}^\infty a^n$ has support the submonoid generated by $X$. We conclude that $\langle X\rangle$ is finite.

Answer 10

Infinitesimal Thurston Rigidity in Holomorphic Dynamics

This concerns eigenvalues of a natural operator on cohomology.

The statement is Galois invariant. The only known proof is not.

Algebraic proofs of algebraic theorems about algebraically closed fields