I wish to show that ideal class group of $R$ is finite. To accomplish this I am given several parts to prove. I have proven some of them but not all. These are the things I am stuck on.
Suppose A is an ideal of $R.$ How do I show that there exists an $\sum_{k = 0}^{p - 2} a_k \varepsilon^k \in A$, where epsilon is the pth root of unity, such that each $|a_k| \leq (N(A))^{\frac{1}{p - 1}}$. where N(A) is the norm of the ideal. I have shown that $N(\sum_{k = 0}^{p - 2} a_k\varepsilon^k) \leq (\sum_{k = 0}^{p - 2} a_k\varepsilon^k)^{p - 1}$, but I am not sure how to use this. Any help will be appreciated.
Here is a summary of what I have done so far:
(1) I realize that N(A)=|R/A|, which is the number of elements in the abelian group R/A. Then this simplifies the goal to finding $a_k$'s such that $|a_k|^{p−1}$ ≤ number of elements in R/A.
(2) $N(A)$ can also be visualized as $\frac{\Delta A}{\Delta R}$ where the numerator and denominator are volumes of the fundamental parallelpiped of A and R respectively.
(3) Norm of cyclotomic integer is a natural number. By the well ordering axiom, for every ideal A, there exists an element $\alpha$ with smallest norm. Perhaps find a lattice basis for A defined as $(\alpha, \beta_1,\beta_2,\ldots,\beta_{p-2})$. Here I used the fact that $A$ and $R$ is a $(p-1)$-dimensional lattice.
Maybe this might help, but I am unsure how to move forward. Any help will be appreciated.