Math 361, Spring 2022, Assignment 15

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

1. Splitting field (of a non-constant polynomial $f\in F[x]$).
2. Isomorphism (of field extensions).
3. Automorphism (of a field extension).
4. $\mathrm{Gal}(F,E,\iota)$ (the Galois group of the extension $(F,E,\iota)$).
5. $\phi(H)$ (the fixed field of the subgroup $H\leq\mathrm{Gal}(F,E,\iota)$).
6. The Galois Correspondence.

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

1. Theorem on existence and uniqueness of splitting fields.
2. Fundamental Theorem of Galois Theory (this is not actually stated in the notes, but you will find a "summary" of the theorem with certain hypotheses left unstated).

Solve the following problems:

1. Using the Sieve of Eratosthenes (or any other suitable method, such as root-searching), show that the polynomial $x^3+x+1$ is irreducible over $\mathbb{Z}_2$.
2. Show that the quotient ring $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$ is a field. (It is usually denoted $GF(8)$.)
3. How many elements does $GF(8)$ have?
4. List the elements of $GF(8)$ explicitly.
5. Define a function $\phi:GF(8)\rightarrow GF(8)$ by the formula $\phi(x)=x^2$. Show that $\phi$ is a unital ring homomorphism. (Hint: to prove that it preserves addition, use the Freshman's Dream.)
6. Compute $\mathrm{ker}(\phi)$.
7. Show that $\phi$ is bijective, and hence an isomorphism from $GF(8)$ to itself. (It is usually called the Frobenius automorphism.)
8. Make a table of values for $\phi$. (This is not as tedious as it appears at first. Remember the Freshman's Dream!)
9. Now define $\iota:\mathbb{Z}_2\rightarrow GF(8)$ by the usual formula $\iota(a)=a+0\alpha+0\alpha^2$, so that $(\mathbb{Z}_2,GF(8),\iota)$ is a field extension. Show that $\phi$ is an automorphism of this extension.
10. It is possible to show that $\phi$ generates the whole of $\mathrm{Gal}(\mathbb{Z}_2,GF(8),\iota)$. Taking this for granted, make a group table for this Galois group.
11. Find all subgroups of $\mathrm{Gal}(\mathbb{Z}_2,GF(8),\iota)$. (Hint: there are very few. Use Lagrange's Theorem!)
12. Compute the Galois Correspondence for $(\mathbb{Z}_2,GF(8),\iota)$.
13. (Optional challenge) Repeat the above exercises for $GF(16)$. (That is, first use the Sieve to identify an irreducible quartic in $\mathbb{Z}_2[x]$, then use this quartic to construct a field with sixteen elements, then make tables for the Frobenius automorphism and its powers, and finally compute the Galois Correspondence. This is no more conceptually challenging than for $GF(8)$, but it is somewhat more tedious. However, $(\mathbb{Z}_2,GF(16),\iota)$ is the smallest field extension for which the Galois group has a non-trivial proper subgroup, so it may be of special interest. Though tedious, this example reveals a number of interesting phenomena.)

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Definitions and Theorems:

https://drive.google.com/file/d/1vFWDTJtosWx0XMEh8ZzTTCTQqueqkB-3/view?usp=sharing

Problems:

https://drive.google.com/file/d/1alSd3CfsFCsDpyeG1-lxmxfFGLz60iIF/view?usp=sharing

Revision Notes:

https://drive.google.com/file/d/1kXciJ1X10AMPXPK3m_ymzX6L7NjYpvO9/view?usp=sharing