# Proving the Kronecker Weber Theorem for Quadratic Extensions

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 8 '16 at 5:48
• Possible duplicate of Square roots of integers and cyclotomic fields – Watson Nov 8 '16 at 7:45
• @Watson: I don't see any "reference material" linked to in your suggestion? – Willie Wong Nov 9 '16 at 1:11
• @WillieWong : I retracted my close vote. – Watson Nov 9 '16 at 15:51

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.