Math 361, Spring 2017, Assignment 13

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

1. Symmetry (of a field extension $F\rightarrow E$).
2. Galois group (of a field extension $F\rightarrow E$).
3. Fixed field (of a subgroup of the Galois group).
4. Group fixing a subextension.
5. Galois correspondence (i.e. the maps $\phi$ and $\gamma$ discussed in class).
6. Frobenius homomorphism.

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

1. The Freshman's Dream.
2. Theorem concerning the splitting field of $x^{\left(p^n\right)}-x\in\mathbb{Z}_p[x]$.
3. Theorem concerning uniqueness of finite fields of a given order.
4. Theorem concerning inclusion-reversal of the Galois correspondence.
5. Fundamental theorem of Galois theory.
6. Theorem concerning the Galois group of $\mathbb{Z}_p\rightarrow GF(p^n)$.

Solve the following problems:

1. Working in $GF(4)$, compute $(1+\alpha)^2$ and $1^2+\alpha^2$. Are these equal? What if the squares are replaced by cubes?
2. Directly compute $x^4$ for each $x\in GF(4)$.
3. Let $E$ denote the splitting field of $x^2-2\in\mathbb{Q}[x]$. Compute the Galois group $\mathrm{Gal}(E/\mathbb{Q})$. Then draw the subgroup diagram of the Galois group and the subextension diagram of $\mathbb{Q}\rightarrow E$, indicating the action of the maps $\phi,\gamma$ on these diagrams.
4. Repeat the problem above, this time with $E$ denoting the splitting field of $x^4-x^2-6\in\mathbb{Q}[x]$.
5. Draw the subextension diagram for $\mathbb{Z}_5\rightarrow GF(15625)$.

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Questions

Galois group of field extension $F \rightarrow E$- the collection of all symmetries of $F \rightarrow E$ is a group under composition. Is the definition right and What would be an example.\

I think I am good with the above$\uparrow$ definition.

It is right. A simple example is $\mathrm{Gal}(\mathbb{C}/\mathbb{R}) = \{\iota,\gamma\}$ where $\iota$ denotes the identity map from $\mathbb{C}$ to itself, and $\gamma$ is the complex-conjugation map $\gamma(z)=\overline{z}$. This is a group of order two, isomorphic to $\mathbb{Z}_2$. -Steven.Jackson (talk) 11:59, 15 May 2017 (EDT)

I have been auguring with a math colleague of mine about the theorem:

The theorem concerning inclusion-reversal of the Galois corresponding: If $H_1 \subseteq H_2$ then $\phi(H_1) \supseteq \phi(H_2)$[$\phi$ is inclusion-reversing]

While my colleague thinks it is

The theorem concerning inclusion-reversal of the Galois corresponding: Suppose $F \rightarrow E$ is any field extension. For any subgroup $H \subseteq Gal(E/F).$

Who is right?

What I had in mind is the first one (if $H_1\subseteq H_2$ then $\phi(H_1)\supseteq\phi(H_2)$) together with the corresponding statement for $\gamma$: If $K_1\subseteq K_2$ then $\gamma(K_1)\supseteq\gamma(K_2)$. -Steven.Jackson (talk) 11:59, 15 May 2017 (EDT)