5 research outputs found
Nontrivial Galois module structure of cyclotomic fields
We say a tame Galois field extension with Galois group has trivial
Galois module structure if the rings of integers have the property that
\Cal{O}_{L} is a free \Cal{O}_{K}[G]-module. The work of Greither,
Replogle, Rubin, and Srivastav shows that for each algebraic number field other
than the rational numbers there will exist infinitely many primes so that
for each there is a tame Galois field extension of degree so that has
nontrivial Galois module structure. However, the proof does not directly yield
specific primes for a given algebraic number field For any
cyclotomic field we find an explicit so that there is a tame degree
extension with nontrivial Galois module structure
A Link to the Math. Connections Between Number Theory and Other Mathematical Topics
Number theory is one of the oldest mathematical areas. This is perhaps one of the reasons why there are many connections between number theory and other areas inside mathematics. This thesis is devoted to some of those connections. In the first part of this thesis I describe known connections between number theory and twelve other areas, namely analysis, sequences, applied mathematics (i.e., probability theory and numerical mathematics), topology, graph theory, linear algebra, geometry, algebra, differential geometry, complex analysis, physics and computer science, and algebraic geometry. We will see that the concepts will not only connect number theory with these areas but also yield connections among themselves. In the second part I present some new results in four topics connecting number theory with computer science, graph theory, algebra, and linear algebra and analysis, respectively. [...] In the next topic I determine the neighbourhood of the neighourhood of vertices in some special graphs. This problem can be formulated with generators of subgroups in abelian groups and is a direct generalization of a corresponding result for cyclic groups. In the third chapter I determine the number of solutions of some linear equations over factor rings of principal ideal domains R. In the case R = Z this can be used to bound sums appearing in the circle method. Lastly I investigate the puzzle “Lights Out” as well as variants of it. Of special interest is the question of complete solvability, i.e., those cases in which all starting boards are solvable. I will use various number theoretical tools to give a criterion for complete solvability depending on the board size modulo 30 and show how this puzzle relates to algebraic number theory