Algebraic Number Fields

Janusz, Gerald J.

doi:10.1090/gsm/007

Cited by 246 publications

(152 citation statements)

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“…Now, condition (3) in the definition of H(d, N, k) implies that the reduction of f (x) − t mod p is cube free. Which, by Kummer's Theorem (see [11,Theorem 7.4], for example), implies that we cannot have pk[α] = P 3 1 P 2 . .…”

Section: Galois Groups Of ϕ N (X) − Tmentioning

confidence: 99%

Iterates of generic polynomials and generic rational functions

Juul¹

2018

Trans. Amer. Math. Soc.

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Abstract. In 1985, Odoni showed that in characteristic 0 the Galois group of the n-th iterate of the generic polynomial with degree d is as large as possible. That is, he showed that this Galois group is the n-th wreath power of the symmetric group S d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.Several of the results proven in this paper were stated and proven by R.W.K. Odoni in an unpublished preprint, including the polynomial versions of the Galois theoretic results and the application presented in Section 6.2. Although the results and arguments given by Odoni in that manuscript are presented somewhat differently here, his work on this project was invaluable in the completion of this paper.

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Section: Galois Groups Of ϕ N (X) − Tmentioning

confidence: 99%

Iterates of generic polynomials and generic rational functions

Juul¹

2018

Trans. Amer. Math. Soc.

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“…The first case occurs when the extended genus field of K equals the genus field of K, and the second when they are different. (See, e.g., [15].) However, our interest in the previous lemma is in the structure of L/K (which is closely related to the genus field and extended genus field of K, but different from both in general).…”

Section: Corollary 8 If CL K Contains a Subgroup Isomorphic To (Z/2z)mentioning

confidence: 99%

Untitled

2011

Proc. Amer. Math. Soc.

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Abstract. Following what is basically Kummer's relatively neglected approach to nonunique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms can be used to explicitly produce all irreducible factorizations of n.

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“…By Kummer's Theorem ( [6], Theorem 7.4), the ideal generated by q in Z[ζ p ] decomposes in exactly the same way as Φ p (X) decomposes in (Z/qZ) [X].…”

Section: Generalized Scholz-type Reciprocity Lawmentioning

confidence: 99%