Let \(p\) be an odd prime and \(F_\alpha\) be the unique cyclotomic extension of degree \(p^{\alpha}\) over \(\Q\text{.}\) Then \(\zeta_{F_\alpha}(-1)\) is a \(p\)-adic integer and
for all nonnegative integers \(\alpha\) and primes \(p\geq5\text{.}\) Here \(\zeta_K(s)\) denotes the Dedekind Zeta function attached to a number field \(K\) (see Chapter 2 for the definition).
Theorem B (Corollary 4.2), as presented in [7, Proposition 2.5] is a consequence of relatively more sophisticated work. A much easier proof, following an approach by John Coates, appeared in Clozel’s paper [6, Proposition 2.4]. The main idea is to use the properties of \(p\)-adic \(L\)–functions. This thesis aims to elucidate the proof presented in [6, Proposition 2.4] and present a refined generalization. To be precise, we prove the following main result of this thesis:
Let \(F_\alpha\) be as above. Then \(\zeta_{F_\alpha}(1-n)\) is a \(p\)-adic integer whenever \(n\) is an even positive integer not divisible by \(p-1\) and
Note that for \(n=2\text{,}\)\(p-1\) does not divide \(n\) for all primes \(p\geq5\text{.}\) Therefore, Theorem 4.1 implies Theorem 4.2 when we take \(n=2\text{.}\)
For the required background knowledge in algebraic number theory and \(p\)-adic \(L\)–functions, we will principally refer to the standard text [47]Introduction to Cyclotomic Fields by Lawrence Washington.
We will also present some new applications of our main result to Milnor \(K\)-groups of number fields. For necessary background in \(K\)-theory, we refer the reader to the excellent self-contained book [15]Handbook of \(K\)-theory. Due to the limited scope of this thesis, we will have to omit many of the technical details.
