Maximal abelian extension ($K^{\mathrm{ab}}$)

The maximal abelian extension $K^{\mathrm{ab}}$ of a field $K$ is the compositum in a fixed separable closure of all finite abelian extensions of $K$.

For a non-Archimedean local field $K$, the Weil group is the inverse image of the infinite cyclic subgroup generated by Frobenius under $\mathrm{Gal}(\overline{K}/K)\to \mathrm{Gal}(\overline{k}/k)\cong \hat{\mathbb{Z}}$. It is dense in the absolute Galois group with its profinite topology.

For a Galois extension $L/K$ of non-Archimedean local fields, the relative Weil group consists of the automorphisms whose residue action is an integral power of Frobenius. It fits intowhen the residue extension is the full algebraic extension.

The Weil-group topology makes the inertia group open with its profinite topology and makes the quotient by inertia discrete. It is generally finer than the topology induced from the ambient Galois group.

The cyclotomic extension ${\mathbb{Q}}_p({\zeta }_{p^n})/{\mathbb{Q}}_p$ is generated by a primitive $p^n$th root of unity. It is totally ramified and has Galois group $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$.

Here $a\ge 0$ and $p\nmid m$. The prime-to-$p$ roots of unity generate an unramified extension of degree equal to the multiplicative order of $p$ modulo $m$; for $m=1$ this degree is one. Indeed, their distinct residues lift uniquely by Hensel lemma, and the Frobenius automorphism acts by raising them to $p$. The $p$-power roots of unity generate a totally ramified extension, because the translated cyclotomic polynomial ${\Phi }_{p^a}(1+T)$ is Eisenstein. The two fields have trivial intersection and their compositum has the same residue degree as the prime-to-$p$ field, by unramified base change preserves total ramification. Thus total ramification is equivalent to $p\equiv 1(\mathrm{mod}m)$.

The local Kronecker-Weber theorem states that every finite abelian extension of ${\mathbb{Q}}_p$ is contained in an extension obtained by adjoining roots of unity.

Local class field theory describes the abelian extensions of a local field $K$ through its multiplicative group $K^{\times }$.

The finite-index open subgroups of $K^{\times }$ are exactly the norm subgroups $N_{L/K}(L^{\times })$ for finite abelian extensions $L/K$. The subgroup determines $L$ uniquely inside $K^{\mathrm{ab}}$.

Local Artin reciprocity gives a continuous homomorphism ${\mathrm{Art}}_K:K^{\times }\to \mathrm{Gal}(K^{\mathrm{ab}}/K)$ with dense image. For every finite abelian extension $L/K$, it induces an isomorphism

For a local field $F$ containing ${\mu }_m$, the symbol is ${\mathrm{rec}}_F(b)(a^{1/m})/a^{1/m}\in {\mu }_m$, after fixing a convention for Local Artin reciprocity. It is independent of the chosen root and bilinear. Its kernel in the second variable is the norm subgroup of $F(a^{1/m})/F$. The quadratic Hilbert symbol is the sign-valued case relevant to quadratic forms.

The symbol is one exactly when $b$ is a norm from $K_v(\sqrt{a})$, and is always one if $a$ is square. Equivalently the conic $z^2=a{x}^2+b{y}^2$ is isotropic. This proves symmetry. The cyclic local norm index makes the norm subgroup have index two for nonsquare $a$, proving bilinearity and nondegeneracy on local square classes. Also $(a,-a{)}_v=1$ and $(a,a{)}_v=(a,-1{)}_v$.

For odd $p$ and p-adic units $u,v$, the quadratic Hilbert symbol isThe Legendre symbols record the two unit square classes. For $p=2$, put $ϵ(u)=(u-1)/2\mathrm{mod}2$ and $\omega (u)=(u^2-1)/8\mathrm{mod}2$. ThenThe basis $-1,2,5$ of the square-class group of the 2-adic numbers gives this formula: its only negative basis pairings are $(-1,-1{)}_2$ and $(2,5{)}_2=(5,2{)}_2$. The first is obstructed by sums of squares modulo four, the second by $x^2-2y^2$ modulo eight; the remaining pairings have explicit field norm representatives.

In the square-class group of a p-adic field, the unit subgroup consists of classes of even valuation. The unramified subgroup consists of the trivial class and the unique nonsquare class defining an unramified extension of degree two. For odd $p$, the two subgroups coincide with $\{1,u\}$ for a nonsquare unit $u$. For $p=2$, they are $\{1,-1,5,-5\}$ and $\{1,5\}$. Norms from an unramified quadratic local field extension have even valuation, and every unit is a norm. The norm characterization of the quadratic Hilbert symbol therefore gives both displayed annihilator identities. Over ${\mathbb{Q}}_2$, the unramified class is $5$ because $(1+\sqrt{5})/2$ satisfies $T^2-T-1$, whose reduction modulo two is irreducible and separable.

For nonzero $a,b$ in a number field, the product of their quadratic Hilbert symbols over all places is one. All but finitely many symbols are one. Over $\mathbb{Q}$, this is equivalent to quadratic reciprocity with its supplementary laws. It supplies the compatibility relation between the local invariants of a global quadratic form.

The local Artin map is the reciprocity homomorphism of Local Artin reciprocity. Its normalization is fixed by choosing whether a uniformizer maps to arithmetic or geometric Frobenius.

For $G=\mathrm{Gal}(L/K)$, Tate cohomology of a finite group identifies the source as ${\hat{H}}^{-2}(G,\mathbb{Z})$ and the target as ${\hat{H}}^0(G,L^{\times })$. Cup product with the local fundamental class is an isomorphism; its inverse is the finite Local Artin map. The algebraic cup-product isomorphism applies because every subgroup has first cohomology zero by Hilbert theorem 90, and second cohomology cyclic of order equal to its size by the local Brauer invariant. Compatibility with restriction and corestriction yields the tower and norm compatibility of Local Artin reciprocity.

With the arithmetic Frobenius normalization, a chosen uniformizer $\pi$ and Lubin–Tate formal group give these actions on the Lubin–Tate tower and the maximal unramified extension. The two fields are linearly disjoint and their compositum is the maximal abelian extension. The inverse on units is essential: using geometric Frobenius would invert the entire Local Artin map. The finite kernels on a compositum of residue degree $d$ and torsion level $r$ are ${\pi }^{d\mathbb{Z}}(1+{\pi }^r{\mathcal{O}}_K)$. They are a basis of finite-index open subgroups, yielding the profinite completion of $K^{\times }$ as the abelian Galois group.

For a finite extension of local fields $L/K$, its norm subgroup is the image of the field norm $N_{L/K}:L^{\times }\to K^{\times }$. For an abelian extension, it is the kernel of the induced local Artin map to $\mathrm{Gal}(L/K)$.

For a finite unramified extension of non-Archimedean local fields, the field norm on residue units is surjective: on a cyclic group of order $q^n-1$ it is exponentiation by $(q^n-1)/(q-1)$ and has image of order $q-1$. For a common uniformizer $\pi$ and $r\ge 1$, expansion of the conjugate product givesThe field trace of a finite finite field extension is onto: its polynomial $X+X^q+\cdots +X^{q^{n-1}}$ is nonzero of degree below $q^n$, so defines a nonzero linear map to the base field. Lift a residue norm and correct successively with elements of $1+{\pi }^r{\mathcal{O}}_L$. The infinite product converges by completeness, and its norm is the prescribed unit. This argument also works in positive characteristic and when $p$ divides $n$.

For a cyclic extension of p-adic fields, the Herbrand quotient of the local multiplicative group together with Hilbert theorem 90 gives this index. If the ramification index and residue degree are $e,f$, the norm valuation formula $v_K(Nx)=f{v}_L(x)$ then gives the unit norm index $e$. In particular all units are norms in an unramified extension.

If $L/K$ is a finite abelian extension of local fields, thenIndeed, Local Artin reciprocity gives total norm index $[L:K]=ef$, while the valuation image of the norm subgroup is $f\mathbb{Z}$.

For a tamely totally ramified extension of degree $q-1$ of a local field with residue field ${\mathbb{F}}_q$, the unit norms are exactly the principal units. Reduction sends a unit norm to its $(q-1)$st power, and the norm-unit index in an abelian local extension shows that this containment has the correct index.

For every positive integer $n$,Under Local Artin reciprocity, the cyclotomic character identifies the quotient with $(\mathbb{Z}/2^n\mathbb{Z}{)}^{\times }$: a uniformizer acts trivially on the totally ramified extension, while a 2-adic unit acts through its residue class modulo $2^n$.