Displaying similar documents to “Patterns and periodicity in a family of resultants”

Kloosterman sums for prime powers in -adic fields

Stanley J. Gurak (2009)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let K be a field of degree n over Q p , the field of rational p -adic numbers, say with residue degree f , ramification index e and differential exponent d . Let O be the ring of integers of K and P its unique prime ideal. The trace and norm maps for K / Q p are denoted T r and N , respectively. Fix q = p r , a power of a prime p , and let η be a numerical character defined modulo q and of order o ( η ) . The character η extends to the ring of p -adic integers p in the natural way; namely η ( u ) = η ( u ˜ ) , where u ˜ denotes the residue...

Cohomology of integer matrices and local-global divisibility on the torus

Marco Illengo (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let p 2 be a prime and let  G be a p -group of matrices in SL n ( ) , for some integer  n . In this paper we show that, when n < 3 ( p - 1 ) , a certain subgroup of the cohomology group H 1 ( G , 𝔽 p n ) is trivial. We also show that this statement can be false when n 3 ( p - 1 ) . Together with a result of Dvornicich and Zannier (see []), we obtain that any algebraic torus of dimension n < 3 ( p - 1 ) enjoys a local-global principle on divisibility by  p .

A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t - 1 and K 2 t of ( ζ p ) , of degrees 2 t - 1 and 2 t over , respectively. The proof requires a particular choice of primitive element for K 2 t over K 2 t - 1 and is based upon the splitting of the cyclotomic polynomial Φ p ( x ) over the subfields.

Powers and alternative laws

Nicholas Ormes, Petr Vojtěchovský (2007)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A groupoid is alternative if it satisfies the alternative laws x ( x y ) = ( x x ) y and x ( y y ) = ( x y ) y . These laws induce four partial maps on + × + ( r , s ) ( 2 r , s - r ) , ( r - s , 2 s ) , ( r / 2 , s + r / 2 ) , ( r + s / 2 , s / 2 ) , that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that n th powers in a free alternative groupoid on one generator are well-defined if and only if n 5 . We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses. ...

On the trace of the ring of integers of an abelian number field

Kurt Girstmair (1992)

Acta Arithmetica

Similarity:

Let K, L be algebraic number fields with K ⊆ L, and O K , O L their respective rings of integers. We consider the trace map T = T L / K : L K and the O K -ideal T ( O L ) O K . By I(L/K) we denote the group indexof T ( O L ) in O K (i.e., the norm of T ( O L ) over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of T ( O L ) (Theorem 1)....