# Math 520A Written Assignment No. 5

#### due Monday, May 7, 2007

Directions. This assignment should be typeset. You must explain the reasoning underlying your answers. If you make use of a reference other than class notes, you must properly cite its use.

You may not seek help from others on this assignment.

1. Write your own proofs of the following propositions:

1. Every polynomial of degree $1$ with coefficients in a field is irreducible.

2. A field $F$ admits no non-trivial algebraic extension if and only if every irreducible polynomial with coefficients in $F$ has degree $1$.

2. If $F$ is a field, a polynomial $f\left(t\right)\in F\left[t\right]$ with coefficients in $F$ determines a “polynomial function” $\phi \left(f\right)$ from $F$ to itself that is defined by $\left(\phi \left(f\right)\right)\left(a\right)=f\left(a\right)\phantom{\rule{0.6em}{0ex}}\text{for}\phantom{\rule{0.6em}{0ex}}a\in F\phantom{\rule{0.6em}{0ex}}\text{.}$ If $A$ denotes the $F$-algebra of all functions $F\to F$, $\phi$ is an $F$-algebra homomorphism $F\left[t\right]\to A$. Show the following:

1. $\phi$ is injective if $F$ is an infinite field.

2. $\phi$ is not injective if $F$ is a finite field.

3. $\phi$ is not surjective if $F$ is an infinite field.

4. $\phi$ is surjective if $F$ is a finite field.

3. A primitive element for a field extension $E⁄F$ is an element $\theta \in E$ such that $E=F\left(\theta \right)$. Find primitive elements for $E$ over $\mathbf{Q}$ in the following cases:

1. $E$ is the splitting field over $\mathbf{Q}$ of ${t}^{12}-1$.

2. $E=\mathbf{Q}\left(\sqrt{2},\sqrt{3}\right)$.

4. More on the polynomial ${t}^{4}+1$:

1. Explain why ${t}^{4}+1$ is irreducible in $\mathbf{Q}\left[t\right]$.

2. Show that ${t}^{4}+1$ is not irreducible over $\mathbf{Z}⁄p\mathbf{Z}$ for every prime $p$.

3. Find the group of $\mathbf{Q}$-algebra automorphisms of the field $\mathbf{Q}\left[t\right]⁄\left({t}^{4}+1\right)\mathbf{Q}\left[t\right]\phantom{\rule{0.6em}{0ex}}\text{.}$

5. For each of the following irreducible polynomials with coefficients in $\mathbf{Q}$ determine the Galois group over $\mathbf{Q}$ of its splitting field:

1. ${t}^{3}-4t+2$.

2. ${t}^{3}-3t-1$.

3. ${t}^{4}-2{t}^{2}-1$.

4. ${t}^{4}-4{t}^{2}+2$.

5. ${t}^{4}-10{t}^{2}+1$.