# Reciprocoity laws

This part is mostly from the note *Reciprocity Laws:
Artin-Hilbert* by Parvati Shastri.

## Kummer theory

Let \(n\) be an integer, and \(K\) be a field with characteristic different from \(n\), which contains all the \(n\)-roots of \(1\). Denote them by \(\zeta_n^i\)s respectively, and the group of these by \(\mu_n\).

#### (Kummer extension) Every Galois extension \(L\) of \(K\) has Galois group \(\mathbb{Z}/n\mathbb{Z}\) iff it is in the form of \(K(\alpha^{1/n})\) with \(\alpha^{1/d}\not\in K\) for every proper divisor \(d\) of \(n\).

**proof** If \(L\) is
such an extension, there is an homomorphism \(Gal(L/K)\rightarrow\mathbb{Z}/n\mathbb{Z}\)
given by sending the automotphism which sends \(\alpha^{1/n}\) to \(\zeta_n^r\alpha^{1/n}\) to \(r\in\mathbb{Z}/n\mathbb{Z}\), here \(\alpha^{1/n}\) is a chosen root of \(x^n-\alpha=0\). It’s easy to see this is an
isomorphism (if not, consider the orbit of roots of \(x^n-\alpha=0\)).

The inverse statement is more interesting. Let \(L/K\) admits a galois group equal to \(\mathbb{Z}/n\mathbb{Z}\) with \(g\) mapping to \(1\). We construct a map \(\phi:Gal(L/K)\rightarrow L^\times\) which sends \(rg\) to \(\zeta_n^r\). View \(\phi\) as an element of \(C^1(Gal(L/K),L^\times)\) and calculation shows that \(d\phi=0\), then \(\phi\) comes from \(dC^0(Gal(L/K),L^\times)\) or to say there’s a \(t\in L^\times\) such that \((rg\cdot t)/t=\phi(rg)=\zeta_n^r\). In particular \(t^n\) is invariant under \(Gal(L/K)\), so now clearly \(t\) is our generating element. \(\blacksquare\)

For such a field, we have a pairing \[ \chi:Gal(\bar{K}/K)\times K^\times\rightarrow\mu_n. \] Namely, given \(\sigma\in Gal(\bar{K}/K)\) and \(x\in K^\times\), define \(\chi_\sigma(x)=(\sigma\cdot y)/y\) where \(y\) is a \(n\)th root of \(x\). Note that this symbol does not depend on the choice of \(y\) since \(\sigma\cdot\zeta_n^i=\zeta_n^i\). If \(x\in K^{\times n}\) then \(\chi_\sigma(x)=1\), so \(\chi\) factors through \(K^\times/K^{\times n}\). Further we have

#### The
Kummer pairing induces an isomorphism \[
K^\times/K^{\times n}\rightarrow Hom(Gal(\bar{K}/K),\mu_n) \]
where \(Hom\) is considered as
morphisms between *continuous* morphisms between topological
groups.

Since \(\mu_n\) is discrete, this
means that every homomorphism in \(Hom(Gal(\bar{K}/K),\mu_n)\) remains zero on
\(Gal(\bar{K}/L)\) for some \(L/K\) finite, or to say the \(Hom\) set is equal to the colimit of \(Hom(Gal(L/K),\mu_n)\).
**proof** Given \(y\in
K^\times/K^{\times n}\), \(K(y^{1/n})/K\) is a nontrivial Galois
extension, so there is an element, say \(g\in
Gal(K(y^{1/n})/K)\) that doesn’t preserve \(y\). That is, \(\chi_g(y)=(g\cdot y)/y\not=1\). This
suggests that the homomorphism is injective. On the other hand, say
\(f:Gal(\bar{K}/K)\rightarrow\mathbb{Z}/n\mathbb{Z}\)
is such a morphism with image group \(\mathbb{Z}/d\mathbb{Z}\), denote by \(H\) the kernel of \(f\) by \(L\) the fixed Galois extension of \(H\), we see that \(Gal(L/K)=\mathbb{Z}/d\mathbb{Z}\) and by
Kummer theory \(L=K(y^{1/d})\) for some
\(y\in K\).

All the morphisms from \(Gal(K(y^{1/d})/K)\) to \(\mu_n\) are linearly dependent, so \(f\) is some \(\chi_\cdot(y^{m/d})\). \(\blacksquare\)

In spite of this *absolute* case, we have another
formulation

#### (Relative case) Subgroups \(\Delta\) of \(K^\times/K^{\times n}\) correspondes bijectively to \(Hom(Gal(K(\Delta^{1/n})/K),\mu_n)\). If \(\Delta\) is infinite then the \(Hom\) is the set of continuous morphisms.

## Local reciprocity law

Local class field theory stats that there’s a functorial isomorphism

for finite abelian extensions of local fields \(L/K\) and \(L'/K'\) where \(L\subset L'\) and \(K\subset K'\) are Galois. We now try to investigate a little more about the morphism \(r_{L/K}\).