If $p$ is a prime and $a^2 ≡ 1$(mod $p$), prove that $a ≡ ± 1 $(mod $p$)

Question

If $p$ is a prime and $a^2 ≡ 1$(mod $p$), prove that $a ≡ ± 1 $(mod $p$)

• We are currently studying modular arithmetic and congruence and I came across this proof on my study set that I'm really not sure how to approach. We have gone over equivalence relations in mod arithmetic, equivalence classes (complete set of residues), and have gone through several mod arithmetic examples w/o variables. Looking through my notes and book I'm still not sure how to proceed with this problem, any help is appreciated.

Answer 1

This is actually a very classical result. The only square roots of $1$ modulo a prime are $+1$ and $-1$. The proof is easy, and I'll give you the beginning. Suppose that $a^2 \equiv 1 \mod p$ where $p$ is an odd prime (for the case $p = 2$, it's trivial). $$a^2 - 1 \equiv 0 \mod p$$ $$(a - 1)(a+1) \equiv 0 \mod p$$

Thus, $p | (a-1)$ or $p | (a+1)$, but not both because $\text{gcd}(a - 1, a + 1)$ is always at most $2$.
Take the needed cases.

$\textbf{Edit}$
And yes, as someone mentioned in the comments, you can always use the fact that an $n$-degree polynomial has at most $n$ roots in a field, assuming you're allowed to use theorems from Abstract Algebra.

Answer 2

1. If $D$ is an integral domain, then $a^{2} =1$ has exactly two solutions: $a^{2} =1 \iff a^{2} - 1 = 0 \iff (a+1)(a-1) = 0$ and since $D$ does not have zero divisors, we have $a = \pm 1$ as the only possible solutions. In relation to your problem, $D = \mathbb{Z}_{p}$ is known to be an integral domain.

2. Also, the hypothesis implies that $a \in G = \mathbb{Z}_{p-1}$ - a cyclic group and so if $a^{2} = 1 = b^{2}$ and $a$ and $b$ are non-identity elements, then $G$ would have two distinct subgroups of the same order, which is not possible in a cyclic group.