Do elements of a finite field extension of degree 2 with norm 1 form a group?

Question

Let $p$ be a prime number. I want to show that elements $\alpha\in\mathbb{F}_{p^2}$ such that $N_{\mathbb{F}_{p^2}\ /\ \mathbb{F}_p}(\alpha)=1$ form a group. I tried to prove this fact on several examples, and my hypothesis is that group has an order $p+1$. Can you help me to prove it or prove the opposite if it's false.

Answer 1

Hint: The norm is a homomorphism $\mathbb{F}_{p^2}^\times\to\mathbb{F}_p^\times$, and you are just considering its kernel.

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The kernel of any group homomorphism is a subgroup, so your set is a group (under multiplication). Since $|\mathbb{F}_{p^2}^\times|=p^2-1$ and $|\mathbb{F}_p^\times|=p-1$, the kernel has at least $\frac{p^2-1}{p-1}=p+1$ elements. On the other hand, the norm of $\alpha$ is just $\alpha\cdot\alpha^p=\alpha^{p+1}$. Since there are at most $p+1$ roots of the polynomial $x^{p+1}-1$ in $\mathbb{F}_{p^2}$, the kernel has at most $p+1$ elements.