Proving the Kronecker Weber Theorem for Quadratic Extensions

Question

I am starting to read about the Kronecker-Weber Theorem. It says that any abelian extension of $\mathbb{Q}$ is contained in a cyclotomic extension. I read somewhere that for quadratic extensions the proof is not very difficult.

Can anyone tell me any reference material for the proof of the kronecker weber theorem for quadratic extensions ?

For a prime $p$ the field $\Bbb{Q}(\zeta_p)$ contains $\sqrt{p^}$, where $p^=(-1)^{(p-1)/2}p$. This gets you started. For example you always find $\sqrt p$ inside $\Bbb{Q}(\zeta_p,i)=\Bbb{Q}(\zeta_{4p})$. With composites, just compose the fields. — Jyrki Lahtonen, Nov 08 '16 at 05:48
Possible duplicate of Square roots of integers and cyclotomic fields — Watson, Nov 08 '16 at 07:45
@Watson: I don't see any "reference material" linked to in your suggestion? — Willie Wong, Nov 09 '16 at 01:11

score 5 · Answer 1 · answered Nov 08 '16 at 06:07

Look up Gauss sums. They express $\sqrt{p}$ as a function of roots of unity. They occur in Gauss's fourth proof of quadratic reciprocity. The classic textbook by Ireland and Rosen "A classical introduction to modern number theory" has extensive discussion of them and their generalisations. Gauss sums are really just Lagrange resolvents for the cyclotomic equation. Lang's book "Algebraic Number Theory" also explains conceptually the relation between reciprocity and the fact that quadratic fields are subfields of cyclotomic fields. Weber's original proof used this as one of his cases.

Proving the Kronecker Weber Theorem for Quadratic Extensions

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