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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
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    Thanks for the info, I've edited that into the question.2012-06-02

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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.