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1

On relative integral bases for unramified extensions

100%

Hutchinson K.

Acta Arithmetica

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1995

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tom 70

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nr 3

279-286

EN

0. Introduction. Since ℤ is a principal ideal domain, every finitely generated torsion-free ℤ-module has a finite ℤ-basis; in particular, any fractional ideal in a number field has an "integral basis". However, if K is an arbitrary number field the ring of integers, A, of K is a Dedekind domain but not necessarily a principal ideal domain. If L/K is a finite extension of number fields, then the fractional ideals of L are finitely generated and torsion-free (or, equivalently, finitely generated and projective) as A-modules, but not necessarily free. Beginning with some classical results of Artin and Chevalley (Propositions 1.1 and 1.2), we give some criteria for the existence or nonexistence of A-bases for ideals in L or for the ring of integers of L in the case where L/K is unramified (Theorem 1.10 and Corollary 2.3). In particular, we show how the existence of an integral basis is (under mild hypotheses) determined by purely group-theoretic properties of the Galois group of the normal closure of L/K. We prove the main results for arbitrary finite separable field extensions L/K. The arguments were suggested by reading [4].

2

Conditions under which $K₂(𝓞_F)$ is not generated by Dennis-Stein symbols

100%

Hutchinson K.

Acta Arithmetica

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1999

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tom 89

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nr 2

189-199

3

Hilbert symbols as maps of functors

63%

Hutchinson K., Ryan D.

Acta Arithmetica

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2004

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tom 114

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nr 4

349-368

Ograniczanie wyników

3 Acta Arithmetica

3 Hutchinson K.

1 Ryan D.

1 2004

1 1999

1 1995

On relative integral bases for unramified extensions

Conditions under which $K₂(𝓞_F)$ is not generated by Dennis-Stein symbols

Hilbert symbols as maps of functors