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searching for Abelian extension 13 found (56 total)

Class field theory (2,212 words) [view diff] exact match in snippet view article find links to article

CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism θ L / F : C F / N L /

Hilbert class field (920 words) [view diff] exact match in snippet view article find links to article

the Hilbert class field of K being K itself: every proper finite abelian extension of K must ramify at some place, and in the extension K(i)/K there

Shafarevich–Weil theorem (416 words) [view diff] exact match in snippet view article find links to article

that F is a global field, K is a normal extension of F, and L is an abelian extension of K. Then the Galois group Gal(L/F) is an extension of the group

Symbol (number theory) (883 words) [view diff] exact match in snippet view article

field K, and takes values in the abelianization of Gal(L/K) for L an abelian extension of K. When α is in the idele group the symbol is sometimes called

Herbrand–Ribet theorem (754 words) [view diff] case mismatch in snippet view article find links to article

ISBN 0-387-94762-0. Mazur, Barry & Wiles, Andrew (1984). "Class Fields of Abelian Extension of Q {\displaystyle \mathbb {Q} } ". Inv. Math. 76 (2): 179–330. Bibcode:1984InMat

Principal ideal (1,332 words) [view diff] exact match in snippet view article find links to article

class field of R {\displaystyle R} ; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction

Ken Ribet (837 words) [view diff] case mismatch in snippet view article find links to article

ISBN 0-387-94762-0. Mazur, Barry & Wiles, Andrew (1984). "Class Fields of Abelian Extension of Q {\displaystyle \mathbb {Q} } ". Inv. Math. 76 (2): 179–330. Bibcode:1984InMat

Greenberg's conjectures (992 words) [view diff] exact match in snippet view article find links to article

theorem) that μ ℓ ( k ) = 0 {\displaystyle \mu _{\ell }(k)=0} for any abelian extension k {\displaystyle k} of the rational number field Q {\displaystyle

Frobenius endomorphism (4,337 words) [view diff] exact match in snippet view article find links to article

easily accomplished if p-adic results will suffice. If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends

Ramification group (2,553 words) [view diff] exact match in snippet view article find links to article

ramification groups for finite subextensions. The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if G

Reciprocity law (1,830 words) [view diff] exact match in snippet view article find links to article

the local reciprocity law. One form of it states that for a finite abelian extension of L/K of local fields, the Artin map is an isomorphism from K × /

Profinite integer (2,090 words) [view diff] exact match in snippet view article find links to article

analogous statement for local class field theory since every finite abelian extension of K / Q p {\displaystyle K/\mathbb {Q} _{p}} is induced from a finite

Riemann hypothesis (16,743 words) [view diff] exact match in snippet view article find links to article

functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis