Does there necessarily exist an element $b$ of the GF(p) satisfying $b^2 = a$?

Question

The question is given below:

All I know about this field id that its order is even $p-1$. And I think if we are speaking about an element other than the identity then this is in general not true (I tried the group $\mathbb{Z_{3}}$ with the addition operation.)

My questions are:

1- Is the question with respect to multiplication operation?

2- should $a$ and $b$ be distinct?

Note 1:

This question was in a linear algebra book.

Note 2:

I have found the answer of this question here Does the element exist in the Galois Field? but I did not understand it and it did not answer the questions I asked above. Also, I think that the answer if the question has answer no should be given by a counterexample, not by proof as I can see in the answer to the mentioned link.

Finally, could anyone help me with that question, please?

Answer 1

You are asking if $a$ is a quadratic residue $\bmod p$. If $p=2$, yes.

If $p$ is odd, not always. See Euler's criterion here.