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Use the Chinese remainder theorem to construct infinitely many $f_i$ such that $f_i$ is Eisenstein at one prime, and $f_i+g$ is Eisenstein at another.

Here are some partial results. I hope that they will help narrow down the search for counterexamples. Suppose that $pq^2=n|\sigma_2(n)=1+p^2+q^2+p^2q^2+q^4$. Claim 1: We must have $q\equiv 1\pmod 4$. …

All constructions of pairs of arithmetically equivalent number fields arise in the following way: start with a Galois extension $F/M$ with Galois group $G$, let $H$, $H'$ be two subgroups that give ri …

You might be interested in this article on Goldbach over function fields. The approach is rather geometric/algebraic, so it does pass your "steers away from hard analysis" test.

The answer to the question as stated is "no". The original Cohen—Lenstra—Martinet heuristics say nothing about the $2$-Sylow subgroup of the class group of a quadratic field. These heuristics were lat …

Each of the quotients in your expression is 0, so yes, there is an isomorphism, but it's between trivial groups. Here is the proof: for a Galois extension $K/\mathbb{Q}$, write down the short exact se …

The general heuristic goes as follows (see the original paper by Cohen-Martinet, but I am being a bit more conservative, since some primes that Cohen-Lenstra-Martinet called "good" seem to not be all …

No, not necessarily. For example, you can take $K\neq \mathbb{Q}$ to be totally real, and take $L$ to be a quadratic CM extension of $K$ whose unit group is equal to that of $K$ (e.g. make sure that $ …

No, the set $S_f$ is empty for lots of forms. For example, if $a$ is not a square modulo $b$, then $ax^2+by^2$ is never a square (consider the equation modulo $b$). Thus, $S_f$ is empty for $f(x,y)=6x …

As stated, this is false: take $L$ and $M$ to be two distinct intermediate cubic subfields of a Galois $S_3$-extension. In particular, they are isomorphic. If a prime of $K$ splits completely in one, …

Corollary 3.17 in this paper of Stefan Keil uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with …

As I have written in your question on SE, if you want to know how to actually compute polynomials that give you ring class fields for a given modulus, then Cohen's Advanced Topics in Computational Num …

The answer to the first question is "yes". See this paper of the Dokchitser brothers, Lemma 3.1 for the case where $K/\mathbb{Q}_p$ is Galois. In the general case, apply the result to the Galois closu …

In general, it does not seem to be any easier to compare two elliptic curves than to say something about each individual curve. In other words, the only results in the direction that you are asking ab …

Such $X$ do indeed exist, and are explicitly constructed in I.Z. Ruzsa, Solving a linear equation in a set of integers, Acta Arith., LXV.3 (1993), pp. 259-282, Theorem 7.5. The whole paper is devoted …