Intereting Posts

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Question on an isomorphism in the proof that $k \cong k \otimes_k k$

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?

- Atiyah - Macdonald Exericse 9.7 via Localization
- Canonical basis of an ideal of a quadratic order
- $L$-function, easiest way to see the following sum?
- How to calculate the ideal class group of a quadratic number field?
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- Solutions to $y^2 = x^3 + k$?
- Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$
- Finite Galois extensions of the form $\frac{\mathbb Z_p}{\langle p(x)\rangle}:\mathbb Z_p$
- When is the sum of two squares the sum of two cubes
- The group of roots of unity in the cyclotomic number field of an odd prime order
- The algebraic closure of a finite field and its Galois group
- Examples where $H\ne \mathrm{Aut}(E/E^H)$
- Value of cyclotomic polynomial evaluated at 1
- How to solve fifth-degree equations by elliptic functions?
- There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

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

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