Exercise 9.2.12

Proof.

(a)

As $p\wedge m=1$, ${\Phi }_p(x)=x^{p-1}+\cdots +x+1$ is irreducible over $\mathbb{Q}({\zeta }_m)$ by Exercise 9.1.16. Therefore the minimal polynomial of ${\zeta }_p$ over $\mathbb{Q}({\zeta }_m)$ is ${\Phi }_p(x)$, of degree $p-1$, so $1,{\zeta }_p,{\zeta }_p^2,\cdots ,{\zeta }_p^{p-2}$ are linearly independent over $\mathbb{Q}({\zeta }_m)$.

(b)

If $a_1,\cdots ,a_{p-1}\in \mathbb{Q}({\zeta }_m)$, as ${\zeta }_p\ne 0$, $$a_1{\zeta }_p+a_2{\zeta }_p^2+\cdots +a_{p-1}{\zeta }_p^{p-1}=0\Rightarrow a_1+a_2{\zeta }_p+\cdots +a_{p-1}{\zeta }_p^{p-1}=0\Rightarrow a_1=a_2=\cdots =a_{p-1}=0,$$

so ${\zeta }_p,{\zeta }_p^2,\cdots ,{\zeta }_p^{p-1}$ are linearly independent over $\mathbb{Q}({\zeta }_m)$.

(c)

Suppose that $\underset{i=1}{\overset{e}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_i(f,{\lambda }_i)=0$, where $a_i\in \mathbb{Q}({\zeta }_m)$. Let $\{[{\lambda }_1],\cdots ,[{\lambda }_e]\}$ be a complete system of representatives of the cosets $[\lambda ]H_f$, then $$\underset{i=1}{\overset{e}{\sum }}{a}_i\underset{a\in [{\lambda }_i]H_f}{\sum }{\zeta }_p^a=0.$$

As ${({\lambda }_i{H}_f)}_{1\le i\le e}$ is a partition of ${(\mathbb{Z}∕\mathit{p\mathbb{Z}})}^{\text{∗}}$, this equality is equivalent to

$$\underset{[k]\in {(\mathbb{Z}∕\mathit{p\mathbb{Z}})}^{\text{∗}}}{\sum }{b}_k{\zeta }_p^k=\underset{k=0}{\overset{p-1}{\sum }}{b}_k{\zeta }_p^k=0,$$

where $b_k$ is a constant on every coset $[{\lambda }_i]H_f$, equal to $a_i$.

Since ${\zeta }_p,{\zeta }_p^2,\cdots ,{\zeta }_p^{p-1}$ are linearly independent over $\mathbb{Q}({\zeta }_m)$, all the $b_k$ are zero, so $a_1=\cdots =a_e=0$.

Thus $f$-periods are linearly independent over $\mathbb{Q}({\zeta }_m)$.