Displaying similar documents to “Cohomology of integer matrices and local-global divisibility on the torus”

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

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Given a monic degree N polynomial f ( x ) ∈ ℤ [ x ] and a non-negative integer ℓ , we may form a new monic degree N polynomial f ℓ ( x ) ∈ ℤ [ x ] by raising each root of f to the ℓ th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) ↦ Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given. ...

A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

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We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t - 1 and K 2 t of ℚ ( ζ p ) , of degrees 2 t - 1 and 2 t over ℚ , 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.

Kloosterman sums for prime powers in -adic fields

Stanley J. Gurak (2009)

Journal de Théorie des Nombres de Bordeaux

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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 ℤ p in the natural way; namely η ( u ) = η ( u ˜ ) , where u ˜ denotes the residue...

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

Kurt Girstmair (1992)

Acta Arithmetica

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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)....

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

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W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers 2 k · 509203 (k = 1, 2,...) is of the form n - φ(n).

On the Number of Partitions of an Integer in the m -bonacci Base

Marcia Edson, Luca Q. Zamboni (2006)

Annales de l’institut Fourier

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For each m ≥ 2 , we consider the m -bonacci numbers defined by F k = 2 k for 0 ≤ k ≤ m - 1 and F k = F k - 1 + F k - 2 + ⋯ + F k - m for k ≥ m . When m = 2 , these are the usual Fibonacci numbers. Every positive integer n may be expressed as a sum of distinct m -bonacci numbers in one or more different ways. Let R m ( n ) be the number of partitions of n as a sum of distinct m -bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for R m ( n ) involving sums of binomial coefficients modulo 2 . In addition we show that this formula may be used to determine the...