Wildly ramified extensions of prime degree &lt;i&gt;p&lt;/i&gt; and Stickelberger conditions

Miyata, Yoshimasa

doi:10.4099/math1924.15.157

Cited by 4 publications

(4 citation statements)

References 5 publications

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“…Now assume that p is odd, and let Σ be the element Σ g∈Cp g in the group ring K[C p ]. For any wildly ramified C p -Galois extension L/K, there is an integral [21] associates to O L a class cl(O L ) in the locally free class group Cl(A L/K ), and then investigates the behaviour of this class as L varies over extensions with the same associated order. Note however that in general O L need not be locally free over A L/K , so cl(O L ) should not be interpreted simply as "the class of" the A L/K -module O L .…”

Section: Realisable Classes and The Proof Of Proposition 18mentioning

confidence: 99%

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On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

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Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O L is free as an O K [G]-module. If O L is free over the associated order A L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G = C p has prime order. We give examples with G = C p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

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Section: Realisable Classes and The Proof Of Proposition 18mentioning

confidence: 99%

“…Miyata [21] has investigated the integral Galois module structure of wildly ramified extensions L/K of number fields of prime degree. A careful reading of his paper suggests that one should expect the O K -ideal Tr L/K (O L ) to be a global obstruction to the freeness of O L over A L/K .…”

Section: Introductionmentioning

confidence: 99%

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

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show abstract

“…PROOF. We use similar arguments as in the proof of [8,Theorem 3]. Let m be an ideal of o'G as in (11) and L(p) be a A,-module in k~G as in (4) and (6); Px=_1..a/tra and (PK)~=L(h(p))a/tr a for pp.…”

Section: Proof For Integers I and J Withmentioning

confidence: 99%

“…Conner and r(c) is the remainder of c on dividing, by p. Let c0=r(c). As in the previous paper [8], we take an integer a of K with (1) where vale, denotes the valuation of Kk. Then, for 0<_i<p,…”

mentioning

confidence: 99%