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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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Thank you very much Zev. –  Bunuelian Trick Jun 2 '12 at 8:05
    
Glad to help. For the first problem, are you sure that you have written the options correctly? Because one cannot conclude any of those statements from the information given. Perhaps there is something about $a$? –  Zev Chonoles Jun 2 '12 at 8:07
    
Zev it will be "for every $a\in \mathbb{F}$, and yes I have written exactly the question i am seeing in a paper. –  Bunuelian Trick Jun 2 '12 at 8:08
    
Thanks for the info, I've edited that into the question. –  Zev Chonoles Jun 2 '12 at 8:09
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1 Answer

up vote 3 down vote accepted

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.

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