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Another approach would be to take the gcd of $N$ and a large power $p^k$ of $p$. This would give $n_1$. In a worst case scenario, $k$ could be $\lg N$, but usually you wouldn't need anything this big. …

For a start the accepted usage for "rational canonical form" in the literature is for a diagonal sum $C(f_1)\oplus C(f_2)\oplus\cdots\oplus C(f_k)$ where $C(f_i)$ is the companion matrix for a monic p …

It's not hard to prove Waterhouse's theorem that all profinite groups are Galois groups. Note first that each quotient of a Galois group by a normal closed subgroup is a Galois group, and as each pro …

I am looking for a reference for the transformation formulae for the classical theta-functions $$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$ and $$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^ …

One big book on distributions is the first volume of Hormander's The Analysis of Linear Partial Differential Operators. This may not be the easiest book to read, but it is comprehensive and a definiti …

There is a whole industry devoted to this. The basic method is by descent, which is a formalized version of the infinite descent proofs of Fermat and Euler. It helps if there are rational 2-torsion po …