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2016•Rocky Mountain Journal of MathematicsOpen access

Galois $p$-groups and Galois modules

Sunil K. Chebolu, Ján Mináč, Andrew Schultz

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Abstract

The smallest non-abelian $p$-groups play a fundamental role in the theory of Galois $p$-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups--as well as other closely related, larger $p$-groups--occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.

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The smallest non-abelian $p$-groups play a fundamental role in the theory of Galois $p$-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups--as well as other closely related, larger $p$-groups--occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.

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Available abstract

The smallest non-abelian $p$-groups play a fundamental role in the theory of Galois $p$-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups--as well as other closely related, larger $p$-groups--occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.

Key concepts: Galois group, Fundamental theorem of Galois theory, Galois cohomology, Embedding problem, Differential Galois theory, Mathematics, Galois module, Galois extension

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