Date of Award
Let F be the field of fractions of R, a ring of power series with coefficients in some field. Let K/F be a finite Galois extension, and assume the integral closure S of R in K is also a power series ring.
In this paper, we consider a construction from ring theory called a crossed product algebra and an associated function f: G × G →K called a cocycle that is used to define a multiplication operation for the crossed product algebra. Consider a crossed product algebra whose cocycle f takes values in S. We give a proof concerning which of f’s values must be invertible in S when G is dihedral. We apply the results of this proof to create an algorithm, which computes the values in the cocycle table. We consider the question of which values the cocycle f can take and thus learn about the possible multiplicative structures for crossed product algebras arising from dihedral field extensions.
Lee, Kaitlyn, "Crossed Product Algebras over Dihedral Field Extensions" (2017). Undergraduate Honors Thesis Collection. 393.