Math 361, Spring 2014, Assignment 14

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

1. Splitting field (of a nonconstant polynomial $p\in F[x]$).
2. Galois group (of a nonconstant $p\in F[x]$).

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

1. Theorem on existence and uniqueness of splitting fields.
2. Theorem concerning finiteness of the Galois group.

Solve the following problems:

1. Give an example of a domain which is not a UFD.
2. Consider the polynomials $p=x^2-2, q=x^3-2,$ and $r=x^4-2$ in $\mathbb{Q}[x]$. Let $E_1, E_2,$ and $E_3$ be the splitting fields of $p, q,$ and $r$, respectively, over $\mathbb{Q}$. Compute the dimensions $[E_1:\mathbb{Q}], [E_2:\mathbb{Q}],$ and $[E_3:\mathbb{Q}]$. (Hint: follow the construction used to prove existence of splitting fields, and use the Dimension Formula. To compute the dimension of $E_3$, you may find the following fact helpful: $x^4-2 = (x^2 + \sqrt{2})(x^2-\sqrt{2}) = (x^2 + \sqrt{2})(x + \sqrt[4]{2})(x - \sqrt[4]{2})$.)
3. Compute the Galois group of $x^2+1\in\mathbb{R}[x]$.

Questions:

1. I just want to make sure my definition for Galois Group is right. When it is a non-constant polynomial and a field, I believe it is the splitting field of that non-constant polynomial over the field. But there was also an important Galois Group definition with two splitting fields where the group contains all isomophisms making the commutative diagram.

See corrected definition below. --Steven.Jackson (talk) 22:05, 13 May 2014 (EDT)

Solutions:

Definitions

Definition: Splitting field (of a nonconstant polynomial $p\in F[x]$).
Fix a polynomial of strictly positive degree $p \in F[x]$. A splitting field for $p$ is a field extension $F \to E$ for which both of the following are true:

1. p splits over E
2. $E=F(\alpha_1,\alpha_2,\dots,\alpha_n)$,and every $\alpha_i$ is a root of $p$.

Example: Working over $\mathbb{R}$, $\mathbb{C}$ is a splitting field for $x^2+1$.

Definition: Galois Group (of a nonconstant $p\in F[x]$).
Fix a polynomial of strictly positive degree $p \in F[x]$. Define the Galois Group of $p$, denoted $Gal_F(p)$, to be the group of automorphisms of the splitting field of $p$ over $F$ which fix $F$ pointwise.
Example: $Gal_{\mathbb{R}}(x^2+1)$ is the group of automorphisms of $\mathbb{C}$ that fix $\mathbb{R}$ pointwise. This is a group of order two, consisting of the identity map and complex conjugation.

Theorems

1. Existence and Uniqueness of Splitting Fields

   LEt $p$ be a polynomial in $F[x]$, where $F$ is a field. A splitting field of $p$ over $F$ exists, and it is isomorphic to all other splitting fields of $p$ over $F$.

2. Finiteness of Galois Group

   The Galois group of $p$ over $F$ is finite, and is a subgroup of $S_n$, where $n$ is the number of distinct roots of $p$.