On ramification in the compositum of function fields

Anbar, Nurdagül; Tutdere, Seher

doi:10.1007/s00574-009-0026-8

Cited by 11 publications

(17 citation statements)

References 3 publications

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“…Note that (3.13) is true even if P does not totally ramify in L/K, by Proposition 3.8. We remark that [1,Theorem 3.4] provides a partial result on the relative different exponents in a compositum of two Artin-Schreier extensions. When P has distinct different exponents in the two extensions -this corresponds to our case m < Mthen P totally ramifies in L, and the result of [1] agrees with (3.13) and (3.14).…”

Section: It Follows Thatmentioning

confidence: 97%

“…We remark that [1,Theorem 3.4] provides a partial result on the relative different exponents in a compositum of two Artin-Schreier extensions. When P has distinct different exponents in the two extensions -this corresponds to our case m < Mthen P totally ramifies in L, and the result of [1] agrees with (3.13) and (3.14). However, when P has the same different exponent in both extensions, then the result of [1] only specifies a range of possible values for the relative different exponents, whereas (3.13) and (3.14) provide the actual values.…”

Section: It Follows Thatmentioning

confidence: 97%

“…Some of the results in this paper can potentially also be derived using local methods; we are indebted to Rachel Pries and the referee for this observation. A special thanks goes to Henning Stichtenoth for his encouragement and suggestions for improvement, and for making [1] available to us during the time of writing. The second-named author's research was supported by NSERC of Canada.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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The Ramification Groups and Different of a Compositum of Artin–schreier Extensions

Scheidler

2010

Int. J. Number Theory

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Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin-Schreier extensions of K. Then much of the behavior of the degree p n extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M . Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

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Section: It Follows Thatmentioning

confidence: 97%

Section: It Follows Thatmentioning

confidence: 97%

Section: Acknowledgmentsmentioning

confidence: 99%

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The Ramification Groups and Different of a Compositum of Artin–schreier Extensions

Scheidler

2010

Int. J. Number Theory

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“…The elementary abelian extension of Galois group Z p × Z p is investigated in [1] and [22]. It should be mentioned that the idea of utilizing transitivity of differents and Hilbert's Different Formula to investigate the ramification groups is used in [9], where the Hasse-Arf property for elementary abelian extensions of function fields is proved.…”

Section: This Relation Is Close If An Intersection Propertymentioning

confidence: 99%

“…Notice that the strictly increasing property and d i ≡ 0 (mod p) are the only two restrictions on the sequence d i of positive integers. See [1] or [22] for a discussion of the type of the extension L/K. Although an explicit construction is not given there, the extension L/K herein is of the same type as described in those two papers.…”

Section: Chapter V Hasse-arf Propertymentioning

confidence: 99%

The ramification group filtrations of certain function field extensions

Castañeda

2015

Pacific J. Math.

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We investigate the ramification group filtration of a Galois extension of function fields, if the Galois group satisfies a certain intersection property. For finite groups, this property is implied by having only elementary abelian Sylow p-subgroups. Note that such groups could be non-abelian. We show how the problem can be reduced to the totally wild ramified case on a p-extension. Our methodology is based on an intimate relationship between the ramification groups of the field extension and those of all degree p sub-extensions. Not only do we confirm that the Hasse-Arf property holds in this setting, but we also prove that the Hasse-Arf divisibility result is the best possible by explicit calculations of the quotients, which are expressed in terms of the different exponents of all those degree p sub-extensions. v ACKNOWLEDGMENTS

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