Front Matter Notations

Throughout this thesis, the rings of integers, rationals, reals, and complex numbers will be denoted as \(\Z,\Q,\R\text{,}\) and \(\C\text{,}\) respectively. We will fix an algebraic closure \(\overline{\Q}\) of \(\Q\text{.}\) Every ring is unital and commutative unless otherwise stated. For any ring \(R\text{,}\) \(R^\times\) denotes the group of units in \(R\text{,}\) \(R[x]\) denotes the ring of polynomials with coefficients in \(R\text{,}\) \(R\llbracket x\rrbracket\) denotes the ring of formal power series with coefficients in \(R\text{,}\) and \(M_n(R)\) denotes the ring (not necessarily commutative) of \(n\times n\) matrices with entries in \(R\text{.}\) For a group \(G\text{,}\) \(R[G]\) denotes the group ring with coefficients in \(R\text{.}\) We use the algebraists’ notation \(\Z/n\Z\) to denote the additive group of residue classes modulo a natural number \(n\text{.}\) The Euler Totient function \(\varphi(n)\) counts the number of elements in \((\Z/n\Z)^\times\text{.}\) For a prime \(p\text{,}\) \(\Z_p\) denotes the ring of \(p\)-adic integers, and \(\Q_p\) denotes the field of \(p\)-adic numbers. We denote the finite field of characteristic \(p\) with \(q=p^f\) elements, for some \(f\in\Z_{\geq1}\text{,}\) by \(\F_q\text{.}\)