In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem, which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebraic values at algebraic points?

The connection with is that Lindemann proved in 1882 that the transcendental function takes transcendental values at every nonzero algebraic number. Since by Euler’s formula, this proves that , and hence itself, is transcendental. In light of this theorem, it is natural to wonder what if anything is special here about the function and the point .

One thing that’s special about is that if is algebraic and is also algebraic, then both and are algebraic for all , and these numbers are all distinct. So one might be led to speculate that if is a transcendental entire function then there are only finitely many algebraic numbers for which is also algebraic.

Unfortunately, as Hilbert knew, this is completely false. For example, the function is transcendental but it takes the rational value 1 at every integer. In 1886, Weierstrass had given an example of a transcendental entire function that takes rational values at all rational numbers; later, in 1895, Stäckel showed that there is a transcendental entire function that takes rational values at all algebraic points. However, the functions of Weierstrass and Stäckel, are in some sense “pathological”; they have large growth rates and do not occur “in nature”. The challenge is to make this intuitive feeling more precise, and also to distinguish from .

One thing that is special about , which is not shared by any of the other functions mentioned in the previous paragraph, is that it satisfies a linear differential equation with rational coefficients (namely ). The existence of such a (not necessarily linear) differential equation turns out to be the key idea needed to generalize Lindemann’s theorem in a substantial way.

Another fruitful generalization is to rephrase our original question as an unlikely intersection problem: given two algebraically independent entire functions and satisfying suitable hypotheses, can we conclude that there are only finitely many complex numbers such that and are simultaneously algebraic? This generalizes our original question by letting and .

Test your intuition: is the following true or false?

Assertion 1: If is a square matrix over a commutative ring , the rows of are linearly independent over if and only if the columns of are linearly independent over .

(All rings in this post will be nonzero commutative rings with identity.)

And how about the following generalization?

Assertion 2: If is an matrix over a commutative ring , the row rank of (the maximum number of -linearly independent rows) equals the column rank of (the maximum number of -linearly independent columns).

In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’d like to explain how the uniqueness portion of that theorem is actually a special case of a more general result, called Fitting’s Lemma, which holds for arbitrary commutative rings.

We begin by proving that one can characterize the diagonal entries in the Smith Normal Form of a matrix over a PID in an intrinsic way by relating them to the GCD of the minors of for all . Actually, since the GCD isn’t defined for general rings, we will instead consider the ideal generated by the minors (which makes sense for any ring, and is the ideal generated by the GCD in the case of a PID).

I’m teaching Graduate Algebra this semester, and I wanted to record here the proof I gave in class of the (existence part of the) structure theorem for finitely generated modules over a PID. It’s a standard argument, based on the existence of the Smith Normal Form for a matrix with entries in a PID, but it’s surprisingly hard to find a concise and accessible reference.

We assume familiarity with basic definitions in the theory of modules over a (commutative) ring. Our goal is to prove the following:

Theorem: Let be a principal ideal domain and let be a finitely generated -module. Then is isomorphic to a direct sum of cyclic -modules. More precisely, there are a non-negative integer (called the rank of ) and elements (called the invariant factors of ) such that for all and .