2 questions on finite fields

Question

I shall be highly glad if anyone can help me to solve these two problems on finite fields. I am writing in the same post as they are related to the same topic.

1. Let $\mathbb{F}$ be a finite field such that $x^2=a$ has a solution in it for every $a\in\mathbb{F}$. Then
   a) It is of characteristic 2
   b) It must have a square number of elements.
   c) Its order is power of 3
   d) Its order is a prime number.
2. If $|\mathbb{F}|=5^{12}$, then what is the total number of subfields of this field?
   a) 3
   b) 5
   c) 8
   d) 6

Answer 1

For problem 1: Recall that $\mathbb{F}^\times=\mathbb{F}\setminus\{0\}$ forms a group under multiplication. Consider the group homomorphism $s:\mathbb{F}^\times \to\mathbb{F}^\times$ defined by $s(x)=x^2$. The assumption in problem 1 says that this function is surjective. Because $\mathbb{F}$ is finite, it is therefore bijective. What does that mean about the kernel of $s$? What does that tell you about the characteristic of $\mathbb{F}$ (can we have $1\neq-1$)?

For problem 2: What is the Galois group $\mathbb{F}_{5^{12}}/\mathbb{F}_5$? Use the Fundamental Theorem of Galois Theory to conclude how many subfields there are.