[Carl Friedrich Gauss, painted by Christian Albrecht Jensen 1840, as copied by Gottlieb Biermann 1887] The “Law of Quadratic Reciprocity”, which I look at in this post, was first proved by Gauss (1777-1855). The starting point is that it is fairly obvious that only half the numbers \(1,2,\dots,p-1\) can be represented as squares \(\bmod p\). [...]
The exercises and problems were here. Exercises E1. Is \(3x^3+15x+10\) irreducible? Yes. 5 divides the coefficients of \(x^2,x\) and the constant term, but not the coefficient of \(x^3\) and \(5^2\) does not divide the constant term, so the polynomial is irreducible by Eisenstein’s criterion. \(\Box\) E2. Is \(x^2+x+2\) irreducible? Obviously Eisenstein’s criterion does not work [...]
[This post is part of a series on the long-standing conjecture that all finite projective planes have prime power order. See here for a list of the other posts. ] Recall that a “division ring” is a field except that the multiplication is not necessarily commutative. The theorem is that any finite division ring is [...]
[Cardinal Sean Brady, bishop of Armagh and head of the Catholic Church in Ireland] One of the worst of the serial child abusers amongst the Catholic priests was Fr Eugene Greene in Raphoe diocese, which is a peaceful country area at the North-Western extreme of Ireland. It is sometimes known (incorrectly, but more helpfully) as [...]
As a preliminary to a proof of Wedderburn’s Theorem in a later post we look at cyclotomic polynomials. As usual “Exercises” are intended to be undemanding, just a way of making sure you have grasped a new concept. “Problems” are intended to be more challenging, although sometimes they turn out to have simple solutions. First [...]
