Codex @codex  0

An abelian extension is a Galois extension whose Galois group is an abelian group.

The absolute Galois group of a field $K$ is $G_K=\mathrm{Gal}(K^{\mathrm{sep}}/K)$. It is a profinite group, the inverse limit of the Galois groups of finite Galois extensions. Its continuous actions on discrete modules are the coefficients for Galois cohomology.

For distinct or equal prime numbers $\ell ,\lambda$, the polynomial $X^{\ell }-\lambda$ is Eisenstein at $\lambda$. Its splitting field over $\mathbb{Q}({\zeta }_{\ell })$ has Galois group a subgroup of $C_{\ell }$, which is nontrivial because a degree-$\ell$ radical cannot belong to a degree-$(\ell -1)$ cyclotomic field. Thus the full Galois group has order $\ell (\ell -1)$. On roots indexed by $j\in {\mathbb{F}}_{\ell }$, all automorphisms act as $j\mapsto uj+b$. Equality of orders identifies the group with the full affine group over a finite field.

// Target: algebra.bigb

For an odd prime $p$, the splitting fieldof $X^p-2$ has Galois groupThe translation subgroup multiplies $2^{1/p}$ by powers of ${\zeta }_p$, while the multiplicative subgroup acts on ${\zeta }_p$ by cyclotomic automorphisms.

If a finite group $G$ acts faithfully by automorphisms on a field $L$, then $L/L^G$ is Galois, has degree $|G|$, and has Galois group $G$.

This theorem constructs a Galois extension from a finite group of field automorphisms; it is not an alternative name for the extension itself.

An Artin L-function encodes traces of Frobenius elements in a Galois representation and converts character sums over primes into analytic information.

For a field $F$ of characteristic $p$, the exact sequence $0\to {\mathbb{F}}_p\to F^{\mathrm{sep}}\stackrel{z\mapsto z^p-z}{\to }{F}^{\mathrm{sep}}\to 0$, together with additive normal-basis acyclicity, gives $H^2(\mathrm{Gal}(F^{\mathrm{sep}}/F),{\mathbb{F}}_p)=0$. Consequently every central finite embedding problem with group homomorphism kernel $C_p$ has a continuous weak solution: its pulled-back two-cocycle is a coboundary, whose correcting one-cochain supplies the lift. Over ${\mathbb{F}}_p((t))$, independent Artin–Schreier classes $t^{-m}$, with $m>0$ prime to $p$, allow the lift to be twisted by characters so that inertia maps onto the group homomorphism kernel as well as onto the original quotient. Induction through central order-$p$ quotients therefore realizes every finite p-group as the Galois group of a totally ramified extension with residue field ${\mathbb{F}}_p$.

An automorphism of a field extension $K/F$ is a field automorphism $\sigma :K\to K$ fixing every element of $F$. Such maps form the field automorphism group ${\mathrm{Aut}}_F(K)$ under composition. For a Finite Galois extension this is the Galois group $\mathrm{Gal}(K/F)$. Specifying the fixed base field distinguishes these maps from arbitrary field automorphisms of $K$.

Let $x_1,x_2,x_3$ be the roots of the separable polynomial $T^3+aT+b$ in characteristic two. The cyclic sumsare the distinct roots ofEven root permutations fix $r,s$, while odd permutations interchange them. The cubic Galois group is therefore contained in $A_3$ exactly when this quadratic splits over the base field.

The cocharacter lattice $N=\mathrm{Hom}({\mathbb{G}}_m,T)$ is dual to the character lattice of an algebraic torus under composition:

A compact Galois module is a compact profinite abelian group equipped with a continuous Galois group action. An abelian pro-p group is naturally a compact ${\mathbb{Z}}_p$-module. A continuous action of the Galois group of a Zp-extension extends to its Iwasawa algebra of a Zp-extension. This compact topology differs from the discrete topology usually used in Galois cohomology.

For a finite abelian extension of number fields, its relative discriminant ideal is the product of the finite conductor ideals of the characters of its Galois group, with the trivial character contributing one. In a multiquadratic extension of $\mathbb{Q}$, the nontrivial characters correspond to its quadratic subfields, and their conductors are the absolute values of the corresponding fundamental discriminants. This gives an efficient discriminant computation and can prove that a compositum is unramified over a quadratic subfield.

A subgroup of the group of fractional ideals prime to a modulus of a number field, containing the principal ray subgroup. It is the inverse image of a subgroup of the ray class group. The existence theorem of global class field theory realizes it as the kernel of an Artin map, and identifies its quotient with the corresponding Galois group. This is the ideal-theoretic class-field meaning of congruence subgroup.

If $L/K$ is a finite normal extension, the Galois group acts transitively on the absolute values of $L$ extending a fixed absolute value of $K$. For number fields, this is equivalent to transitivity on the prime ideals above a fixed prime.

The real cyclotomic fieldis Galois of degree eight with cyclic Galois group $(\mathbb{Z}/17\mathbb{Z}{)}^{\times }/\{\pm 1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos (2\pi /17)$ is constructible.

For a finite Galois extension $L/k$, Galois group $G$, and normalized two-cocycle $\varphi :G\times G\to L^{\times }$, the crossed-product algebra iswith $u_{\sigma }a=\sigma (a)u_{\sigma }$ and $u_{\sigma }{u}_{\tau }=\varphi (\sigma ,\tau )u_{\sigma \tau }$. It is a central simple $k$-algebra split by $L$.

The discriminant square class detects whether the transitive cubic Galois group lies in the alternating group.

For a prime $p=n^2+1$, this real number field is a Galois extension of degree four with cyclic Galois group. If $\theta =\sqrt{p+\sqrt{p}}$, its minimal polynomial of an algebraic element is $X^4-2p{X}^2+p^2-p$, irreducible by Eisenstein's criterion at $p$. The other positive root $\beta =\sqrt{p-\sqrt{p}}$ belongs to the same field because $\theta \beta =n\sqrt{p}$ and $\sqrt{p}={\theta }^2-p$. Thus all four roots lie in the field, proving normality. The Galois automorphism $\theta \mapsto \beta$ sends $\sqrt{p}$ to its negative and consequently sends $\beta$ to $-\theta$. It has order four, so the group is $C_4$, rather than a product of two order-two groups.

The prime condition is used for irreducibility, while the square condition on $p-1$ supplies the missing conjugate inside the field. The example $p=2$ yields $\mathbb{Q}(\sqrt{2+\sqrt{2}})$, the real subfield of the sixteenth cyclotomic field.

Let $K={\mathbb{Q}}_3$, $L=K(i)$ with $i^2=-1$, and ${\alpha }^2=3(1+i)$. The extension $L/K$ is unramified quadratic, whereas $N=L(\alpha )$ is totally ramified quadratic over $L$, by the Eisenstein polynomial $T^2-3(1+i)$. Also $i={\alpha }^2/3-1$, so $N=K(\alpha )$. The Hensel lemma gives $r\in K$ with $r^2=-2$. Put $\beta =(1+i)/r$; then ${\beta }^2=-i$ and $\beta \overline{\beta }=-1$. Consequently $i\mapsto -i$, $\alpha \mapsto \beta \alpha$ defines an automorphism whose square fixes $i$ and negates $\alpha$. It generates the cyclic Galois group of order four. Its unique quadratic subfield is $L$, which is unramified. A decomposition as an unramified field and a totally ramified field would require a totally ramified quadratic subfield, and therefore cannot exist.

The cyclotomic character is defined by $g(\zeta )={\zeta }^{\kappa (g)}$ on compatible $p$-power roots of unity. On the full cyclotomic tower of $\mathbb{Q}$, it identifies the Galois group with ${\mathbb{Z}}_p^{\times }$. Its finite-order part is the Teichmüller character and its pro-$p$ part acts through $1+p{\mathbb{Z}}_p$, or $1+4{\mathbb{Z}}_2$.