منابع مشابه
Galois Groups as Permutation Groups
Writing f(T ) = (T − r1) · · · (T − rn), the splitting field of f(T ) over K is K(r1, . . . , rn). Each σ in the Galois group of f(T ) over K permutes the ri’s since σ fixes K and therefore f(r) = 0⇒ f(σ(r)) = 0. The automorphism σ is completely determined by its permutation of the ri’s since the ri’s generate the splitting field over K. A permutation of the ri’s can be viewed as a permutation ...
متن کاملFinite Abelian Groups as Galois Groups
The purpose of this note is to record for my algebra class a proof of the Inverse Galois Problem for finite abelian groups. Recall that the Inverse Galois Problem is stated as follows: Given a finite group G, is there a Galois extension Q ⊆ K such Gal(K/Q) = G? The crucial point in the problem is that the base field is Q, since given any finite group G, there is a Galois extension of fields F ⊆...
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By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.
متن کاملGalois as Permutation Groups
Writing f(X) = (X− r1) · · · (X− rn), the splitting field of f(X) over K is K(r1, . . . , rn). Each σ in the Galois group of f(X) over K permutes the ri’s since σ fixes K and therefore f(r) = 0⇒ f(σ(r)) = 0. The automorphism σ is completely determined by its permutation of the ri’s since the ri’s generate the splitting field over K. A permutation of the ri’s can be viewed as a permutation of th...
متن کاملTheta functions on covers of symplectic groups
We study the automorphic theta representation $Theta_{2n}^{(r)}$ on the $r$-fold cover of the symplectic group $Sp_{2n}$. This representation is obtained from the residues of Eisenstein series on this group. If $r$ is odd, $nle r
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 2000
ISSN: 0021-8693
DOI: 10.1006/jabr.1999.8097