# p-adic Numbers and Dissections of Squares into Triangles

My thesis advisor Robert Coleman passed away two years ago today (see this remembrance on my blog).  One of the things I learned from Robert is that p-adic numbers have many unexpected applications (see, for example, this blog post).  Today I want to share one of my favorite surprising applications of p-adic numbers, to a simple problem in Euclidean geometry. Continue reading

# Probability, Primes, and Pi

What is the probability that two randomly chosen integers have no prime factors in common?  In honor of Pi Day, I’d like to explain the surprising answer: $6/\pi^2$.

The hero of this story is Leonhard Euler, who worked out this astonishing connection between prime numbers and $\pi$ through a series of brilliant insights.  In the spirit of Euler, I will be rather cavalier about issues of convergence and rigor here, focusing on the key underlying ideas.

Today marks the birthday of Karl Friedrich Gauss, the “Prince of Mathematicians”, who was born on April 30, 1777.  In honor of Gauss’s 238th birthday, I thought I would blog about one of Gauss’s favorite theorems — the Law of Quadratic Reciprocity — and its relation to the sign of the quadratic Gauss sum, which we will determine using the Discrete Fourier Transform.  Our exposition mostly follows this paper by Ram Murty.  Regarding the sign of the quadratic Gauss sum, Gauss conjectured the correct answer in his diary in May 1801, but it took more than four years until he was able to find a proof in August 1805. Gauss wrote to his friend W. Olbers that seldom had a week passed for four years that he had not tried in vain to prove his conjecture.  Then:

Finally, two days ago, I succeeded – not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.

# A p-adic proof that pi is transcendental

Ferdinand von Lindemann

In my last blog post, I discussed a simple proof of the fact that pi is irrational.  That pi is in fact transcendental was first proved in 1882 by Ferdinand von Lindemann, who showed that if $\alpha$ is a nonzero complex number and $e^\alpha$ is algebraic, then $\alpha$ must be transcendental.  Since $e^{i \pi} = -1$ is algebraic, this suffices to establish the transcendence of $\pi$ (and setting $\alpha = 1$ it shows that $e$ is transcendental as well).  Karl Weierstrass proved an important generalization of Lindemann’s theorem in 1885.

The proof by Lindemann that pi is transcendental is one of the crowning achievements of 19th century mathematics.  In this post, I would like to explain a remarkable 20th century proof of the Lindemann-Weierstrass theorem due to Bezivin and Robba [Annals of Mathematics Vol. 129, No. 1 (Jan. 1989), pp. 151-160], which uses p-adic analysis in a key way.  Their original argument was made substantially more elementary by Beukers in this paper; we refer the reader to [American Mathematical Monthly Vol. 97 Issue 3 (Mar. 1990), pp. 193-197] for a lovely exposition of the resulting proof, which rivals any of the usual approaches in its simplicity.  But I’d like to focus here on the original Bezivin-Robba proof, which deserves to be much better known than it is.  In the concluding remarks, we will briefly discuss a 21st century theorem of Bost and Chambert-Loir that situates the Bezivin-Robba approach within a much broader mathematical framework. Continue reading

# Newton polygons and Galois groups

Issai Schur

A famous result of David Hilbert asserts that there exist irreducible polynomials of every degree $n$ over ${\mathbf Q}$ having the largest possible Galois group $S_n$.  However, Hilbert’s proof, based on his famous irreducibility theorem, is non-constructive.  Issai Schur proved a constructive (and explicit) version of this result: the $n^{\rm th}$ Laguerre polynomial $L_n(x) = \sum_{j=0}^n (-1)^j \binom{n}{j} \frac{x^j}{j!}$ is irreducible and has Galois group $S_n$ over ${\mathbf Q}$.

In this post, we give a simple proof of Schur’s result using the theory of Newton polygons.  The ideas behind this proof are due to Robert Coleman and are taken from his elegant paper On the Galois Groups of the Exponential Taylor Polynomials.  (Thanks to Farshid Hajir for pointing out to me that Coleman’s method applies equally well to the Laguerre polynomials.) Before we begin, here is a quote from Ken Ribet taken from the comments section of this post: