Math 361, Spring 2018, Assignment 13

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

1. Solvable by radicals.
2. Symmetry (of a field extension).
3. Galois group (of a field extension).

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

1. Theorem concerning existence and uniqueness of splitting fields.
2. Theorem concerning $(q-1)$st powers in fields of order $q$.
3. Theorem concerning $q$th powers in fields of order $q$.
4. Theorem characterizing fields of order $q$ as splitting fields.
5. Theorem concerning uniqueness of fields of order $q$.

Solve the following problems:

1. Let $F_1$ denote the quotient ring $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$. Show that $F_1$ is a field with eight elements. For purposes of the problems below, denote the generator of this field by $\alpha$.
2. Let $F_2$ denote the quotient ring $\mathbb{Z}_2[x]/\left\langle x^3+x^2+1\right\rangle$. Show that $F_2$ is a field with eight elements. For purposes of the problems below, denote the generator of this field by $\beta$.
3. Working in $F_2$, find a root of the polynomial $x^3+x+1$.
4. Construct an isomorphism from $F_1$ to $F_2$. (Hint: start with the evaluation homomorphism $\phi_r:\mathbb{Z}_2[x]\rightarrow F_2$, where $r$ is the root you found above, and use the Fundamental Theorem of Homomorphisms.)
5. Make an explicit table of values for the isomorphism you constructed above.
6. Compute the Galois group of the polynomial $x^2-2\in\mathbb{Q}[x]$. (Hint: imitate what we did in class to compute the Galois group of $\mathbb{R}\rightarrow\mathbb{C}$.)

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