$(\mathbb{F}_p^\times)^2$ subgroup of $\mathbb{F}_p^\times$ and $(\mathbb{F}_p^\times)/(\mathbb{F}_p^\times)^2 \cong \mathbb{Z}/ 2\mathbb{Z}$

Question

I was wondering if someone here can help me with the following question:

Let $p$ be a prime number. Show that the set $(\mathbb{F}_p^\times)^2$ of squares in $\mathbb{F}_p^\times$ is a normal subgroup of $\mathbb{F}_p^\times$, and that the quotient group $\mathbb{F}_p^\times/(\mathbb{F}_p^\times)^2$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.

Answer 1

Hint:

• Let $g$ be a generator of $\Bbb F_p^\times$, then $$A=\{1,2,3,\ldots,p-1\}=\{g,g^2,\ldots,g^{p-1}\}=B$$ How many elements in $B$ are squares? Is it possible that $g$ itself is a square?

• For $\left(\Bbb F_p^\times\right)^2$ to be a normal subgroup you need to have $$x\in\left(\Bbb F_p^\times\right)^2\iff gxg^{-1}\in\left(\Bbb F_p^\times\right)^2$$ for every $x\in\left(\Bbb F_p^\times\right)^2$ and $g\in\Bbb F_p^\times$. What is $gxg^{-1}$?

Answer 2

Hint: Consider the map $\mu: \mathbb{F}_p^\times \to \mathbb{F}_p^\times$ given by $\mu(x)=x^2$. Prove that $\mu$ is a homomorphism. Compute its image and its kernel.