Math 361, Spring 2018, Assignment 14

Carefully define the following terms, then give one example and one non-example of each:

1. Conjugate (elements of an extension field).
2. Normal extension.
3. Separable extension.
4. Solvable group.

Carefully state the following theorems (you do not need to prove them):

1. Theorem constraining where an element of the Galois group can send an element of an extension field.
2. Theorem giving sufficient conditions for the bijectivity of the Galois correspondence.
3. The Freshman's Dream.

Solve the following problems:

1. Prove the freshman's dream. (Hint: start with the Binomial Theorem, which is true in any commutative ring. This says that for any $a,b$ in such a ring, and for any positive integer $n$, one has $$(a+b)^n = \sum_{i=0}^n{n\choose i}a^{n-i}b^i = a^n + {n\choose 1}a^{n-1}b+\dots+{n\choose n-1}ab^{n-1}+b^n.$$ Now suppose $n=p$ is prime. Using this fact, together with the unique prime factorization of integers, show that each binomial coefficient ${n\choose i}$ for $1\leq i\leq n-1$ is in fact divisible by $p$, and hence collapses to zero in any field of characteristic $p$.)
2. Suppose $q=p^n$ is some power of $p$, and consider the map $\phi:GF(q)\rightarrow GF(q)$ defined by $\phi(x)=x^p$. Prove that $\phi$ is a field automorphism of $GF(q)$. (In the literature, this map is called the Frobenius automorphism. It can be shown that the Frobenius automorphism generates the Galois group of $GF(q)$, regarded as an extension of $\mathbb{Z}_p$.)
3. Make explicit tables of values for the Frobenius automorphisms of $GF(4)$ and $GF(8)$.

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