A question about quadratic non-residues in finite fields.

Question

Let $p$ be a prime such that $p \equiv 1$ (mod $3$) and $p \equiv 3$ (mod $4$). Consider a quadratic non-residue $b \in \mathbb{F}_{p^2}$. Is $\mathrm{Norm}_{\mathbb{F}_{p^2}/ \mathbb{F}_{p}}(b)$ a quadratic non-residue in $\mathbb{F}_{p}$ under these conditions? Help me please.

Answer 1

The conjugate of $b$ over $\mathbb F_p$ is $b^p$, so $\mathrm{Norm}_{\mathbb{F}_{p^2}/ \mathbb{F}_{p}}(b)=bb^p=b^{p+1}$. By Euler's criterion, it's quadratic character in $\mathbb F_p$ is equal to $(b^{p+1})^{(p-1)/2}=b^{(p^2-1)/2}$, which is equal to the quadratic character of $b$ in $\mathbb F_{p^2}$.

Note: I did not use either of the conditions on $p$ stated in the question, just the fact that $p$ is odd.