Man zeige, dass L=ℚ(√(p) ……√(pn)) für paarweise verschiedene Primzahlen p1 …pn eine abelsche Gal

step 1 In this question, value of m plus nth term of a gp and value of m minus nth term of a gp is give

Man zeige, dass L=ℚ(√(p) ……√(pn)) für paarweise verschiedene...

Man zeige, dass $L=\mathbb{Q}\left(\sqrt{p} \ldots \ldots \sqrt{p_{n}}\right)$ für paarweise verschiedene Primzahlen $p_{1} \ldots p_{n}$ eine abelsche Galois-Erweiterung von $Q$ mit Galois-Gruppe (Z/2Z) $^{n}$ ist. (Anleitung: Man beachte, dass fïr $a \in \mathbb{Q}$ mit $\sqrt{a} \in L$, und für $\sigma \in \operatorname{Cal}(L / Q)$ stets $\sigma(\sqrt{a})=\pm \sqrt{a}$ gilt. In allgemeinerem Rahmen ist dies der Ausgangapunkt der so genannten Kummer-Theork. Fur die von $p_{1}, \ldots, p_{\text {n erzeugte multiplika- }}$ tive Untergruppe $M \subset Q^{*}$ lässt sich dann $M / M^{2}$ als Untergruppe der Gruppe $\operatorname{Hom}(\operatorname{Gal}(L / Q), Z / 2 Z)$ aller Gruppenhomomorphismen $\operatorname{Gal}(L / Q) \longrightarrow Z / 2 Z$ betrachten.)

Key Concepts

Multiplicative Groups Modulo Squares

For a given base field, the subgroup of its multiplicative group generated by a set of elements (such as distinct primes) can be considered modulo squares, forming the group M/M^2. This quotient captures the essential information regarding which elements admit square roots in the extension. Through Kummer theory, this group is linked to the dual group of the Galois group, revealing a correspondence between the algebraic structure of the field and the combinatorial structure of the Galois group.

Kummer Theory

Kummer theory studies abelian extensions of fields that are obtained by adjoining n-th roots of elements, especially under the condition that the base field contains enough roots of unity. In the quadratic case, where n = 2, Kummer theory provides a framework by relating the field extension with the multiplicative structure of the base field, in particular through the use of the quotient group M/M^2 for a suitable multiplicative subgroup M.

Galois Extensions

A Galois extension is a field extension that is both normal and separable. This property ensures that the extension has a well-defined Galois group—that is, the group of field automorphisms that fix the base field—which provides deep insight into the structure of the extension and allows one to study the solvability of polynomials and symmetries within the field.

Quadratic Field Extensions

Quadratic field extensions are obtained by adjoining the square root of an element (typically not a square in the base field) to the base field. In such extensions, each automorphism sends the square root either to itself or its negative, which underlies the binary structure that leads to a Galois group isomorphic to a direct sum of copies of Z/2Z when several independent square roots are adjoined.

Galois Groups and Automorphisms

The Galois group of an extension is the group of all field automorphisms that leave the base field fixed. In the context of extensions obtained by adjoining square roots, each automorphism has the characteristic property of sending each square root to either itself or its negative. This binary action results in an abelian group structure where the overall Galois group is isomorphic to a direct product of multiple copies of Z/2Z, reflecting the independent choices made for each square root.

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