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 $$\left(\phi \left(f\right)\right)\left(a\right)=f\left(a\right)\text{for}a\in F\text{.}$$ 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 $$\mathbf{Q}\left[t\right]⁄\left(t^4+1\right)\mathbf{Q}\left[t\right]\text{.}$$

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. $t^4-2t^2-1$.

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

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