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Displaying similar documents to “Pólya fields, Pólya groups and Pólya extensions: a question of capitulation”

On the maximal unramified pro-2-extension over the cyclotomic ℤ 2 -extension of an imaginary quadratic field

Yasushi Mizusawa (2010)

Journal de Théorie des Nombres de Bordeaux

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For the cyclotomic ℤ 2 -extension k ∞ of an imaginary quadratic field k , we consider the Galois group G ( k ∞ ) of the maximal unramified pro- 2 -extension over k ∞ . In this paper, we give some families of k for which G ( k ∞ ) is a metabelian pro- 2 -group with the explicit presentation, and determine the case that G ( k ∞ ) becomes a nonabelian metacyclic pro- 2 -group. We also calculate Iwasawa theoretically the Galois groups of 2 -class field towers of certain cyclotomic 2 -extensions.

Hilbert-Speiser number fields and Stickelberger ideals

Humio Ichimura (2009)

Journal de Théorie des Nombres de Bordeaux

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Let p be a prime number. We say that a number field F satisfies the condition ( H p n ′ ) when any abelian extension N / F of exponent dividing p n has a normal integral basis with respect to the ring of p -integers. We also say that F satisfies ( H p ∞ ′ ) when it satisfies ( H p n ′ ) for all n ≥ 1 . It is known that the rationals ℚ satisfy ( H p ∞ ′ ) for all prime numbers p . In this paper, we give a simple condition for a number field F to satisfy ( H p n ′ ) in terms of the ideal class group of K = F ( ζ p n ) and a “Stickelberger ideal” associated to the...

Absolute norms of p -primary units

Supriya Pisolkar (2009)

Journal de Théorie des Nombres de Bordeaux

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We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about 2 -primary units. We also prove a similar statement about the absolute norms of p -primary units, for all primes p .

Pólya fields and Pólya numbers

Amandine Leriche (2010)

Actes des rencontres du CIRM

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A number field K , with ring of integers 𝒪 K , is said to be a Pólya field if the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field K is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of K in a Pólya field. We give a positive answer to this embedding problem by...

Markoff numbers and ambiguous classes

Anitha Srinivasan (2009)

Journal de Théorie des Nombres de Bordeaux

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The Markoff conjecture states that given a positive integer c , there is at most one triple ( a , b , c ) of positive integers with a ≤ b ≤ c that satisfies the equation a 2 + b 2 + c 2 = 3 a b c . The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9 c 2 - 4 , every ambiguous form in the principal genus corresponds to a divisor of 3 c - 2 , then the conjecture is true. As a result, we obtain criteria...