Existence of the group of multipicative represents of elements of $\mathbb{F}_p^\ast$

Question

let $U=\mathbb{Z}_p^\ast$ the group of $p$-adic units. And $V=\{x\in U\mid x^{p-1} = 1\}$ the group of multiplicative representations of the elements of $\mathbb{F}_p^\ast$.

It is stated, that the existence of this set follows when you apply the following lemma:

Every simple zero of the reduction modulo $p$ of a polynomial $f$ lifts to a zero of $f$ with coefficients in $\mathbb{Z}_p$.

To the function $f(x)=x^{q-1}-1$.

Going after the lemma we have to solve $x^{q-1}-1\equiv 0\mod p$. Therefore $x^{q-1}\equiv 1\mod p$. Obviously $x=1$ is a solution. We have to check, that $f'(1)\not\equiv 0\mod p$.

$$f'(x)=(q-1)x^{q-2} \text{ and } f'(1)=q-1. \quad q-1\not\equiv 0\bmod p\Longleftrightarrow p\nmid q-1$$

But I do not see why this has to hold. For the existence of $V$ we have to show, that $V\neq\emptyset$, am I right?

How do I have to apply the lemma here? Thanks in advance.

Answer 1

An alternative to using Hensel's lemma. A roadmap to proving that $|V|=p-1$. You fill in the remaining details.

1. Fix an integer $a$ in the range $1,2,\ldots,p-1$.
2. Define the sequence $x_n=a^{p^n}$, $n=1,2,3,\ldots$.
3. Show that whenever $n<m$ we have the congruence $$x_m\equiv x_n\pmod{p^n}.$$
4. Conclude that $|x_m-x_n|_p\le 1/p^n$ whenever $n<m$. Therefore the sequence $(x_n)$ is Cauchy.
5. By completeness of $\Bbb{Q}_p$ we can conclude that the limit $$\zeta_a=\lim_{n\to\infty}x_n$$ exists as an element of $\Bbb{Q}_p$.
6. Show that $x_{n+1}=x_n^p$. Conclude that $\zeta_a^p=\zeta_a$.
7. Show that $\zeta_a\equiv a\pmod p$.
8. Show that $\zeta_a\neq0$ for all $a$.
9. Show that $\zeta_a^{p-1}=1$ for all $a$.
10. Show that $\zeta_a\neq\zeta_b$ whenever $a,b$ are distinct integers in the range $0<a,b<p$. Conclude that you have found $p-1$ distinct zeros of $f(x)=x^{p-1}-1$ in $\Bbb{Q}p$.

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Using the Lemma:

• Show that $a\in\Bbb{F}_p^*$ is a zero of $f(x)=x^{p-1}-1$.
• Show that $f'(a)\neq0.$
• The Lemma says that $f(x)$ has a zero $\zeta_a\in\Bbb{Z}_p$ such that $\zeta_a\equiv a\pmod{p\Bbb{Z}_p}.$
• See item 10 from the first solution.