Intereting Posts

Matrices A+B=AB implies A commutes with B
Is the natural map $L^p(X) \otimes L^p(Y) \to L^p(X \times Y)$ injective?
Proof of Froda's theorem
S4/V4 isomorphic to S3 – Understanding Attached Tables
The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$
$\mathbb Z^n/\langle (a,…,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
Exponential bound on norm of matrix exponential (of linear ODE)
Every infinite Hausdorff space has an infinite discrete subspace
Explanation of the Bounded Convergence Theorem
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
Showing that $\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$
Triangle Formula for alternative Points
$\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ iff $n=m$
Can we use this formula for a certain indeterminate limit $1^{+\infty}$?

Since $\varphi(p)=p-1$ is even the p’th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field.

What is this theorem called and how is it proved?

- Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?
- Intuition regarding Chevalley-Warning Theorem
- The Langlands program for beginners
- Pre-requisites needed for algebraic number theory
- Linear independence of roots over Q
- Proof of Stickelberger’s Theorem

- Galois Group of $(x^3-5)(x^2-3)$
- Quadratic subfield of cyclotomic field
- minimal polynomial of power of algebraic number
- Compositum of abelian Galois extensions is also?
- Show that $\langle 13 \rangle$ is a prime ideal in $\mathbb{Z}$
- Polynomial splitting in linear factors modulo a prime ideal
- surjectivity of group homomorphisms
- Intuition behind looking at permutations of the roots in Galois theory
- How to describe the Galois group of the compositum of all quadratic extensions of Q?
- Powers of $x$ as members of Galois Field and their representation as remainders

This is a special case of the Kronecker-Weber theorem, which says that any abelian extension of $\mathbb{Q}$ is contained inside some cyclotomic field; any quadratic extension of $\mathbb{Q}$ is automatically abelian. I don’t believe the special case of the theorem for quadratic fields has a separate name.

However, one does not need the full power of this (very advanced) theorem. The following two steps are used in exercise 8 of Chapter 2 in Marcus’s *Number Fields* to prove precisely the case of quadratic fields:

**Show that $\mathbb{Q}(\zeta_p)$ contains $\sqrt{p}$ if $p\equiv 1\bmod 4$ and $\sqrt{-p}$ if $p\equiv 3\bmod 4$.**(Note: This step follows from results in the preceding chapter in Marcus about the discriminant of $\mathbb{Q}(\zeta_p)$ being

$$\prod_{1\leq r<s\leq p-1}(\zeta_p^r-\zeta_p^s)^2=\left(\prod_{1\leq r<s\leq p-1}(\zeta_p^r-\zeta_p^s)\right)^2=\pm p^{p-2}$$**Show that $\mathbb{Q}(\zeta_8)$ contains $\sqrt{2}$.**

Obviously $\mathbb{Q}(i)$ contains $\sqrt{-1}$, so to get a cyclotomic field containing $\mathbb{Q}(\sqrt{m})$, we just need to take the compositum of the cyclotomic fields corresponding to each of $m$’s prime factors (note that we can assume $m$ is squarefree).

This result of Gauss is very well-known in algebraic number theory. The obvious web searches will easily locate proofs and much more, e.g. here’s a proof from Weintraub: Galois Theory

- Localizations of $\mathbb{Z}/m\mathbb{Z}$
- Cartesian Product in mathematics
- Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.
- If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.
- Book on combinatorial identities
- How do you integrate Gaussian integral with contour integration method?
- decimal representation of $2^m$ starts with a particular finite sequence of decimal digits
- Are there more rational or irrational numbers?
- Provable Hamiltonian Subclass of Barnette Graphs
- question about solving recurrences
- An inequality of J. Necas
- Finite family of infinite sets / A.C.
- General solution for the Eikonal equation $| \nabla u|^2=1$
- $\sin(x) = \sum{a_n \sin(n \log(x))+b_n \cos(n \log(x))}$
- Is there a similar concept for a sigma algebra like a base for a topology?