Math 520A Written Assignment No. 4

due Monday, April 23, 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. Decompose the polynomial $t^{12}-1\in F\left[t\right]$ as the product of irreducible polynomials when $F$ is the field

   1. $\mathbf{Q}$.

   2. $\mathbf{Z}⁄5\mathbf{Z}$.

2. Let $A$ denote the ring $$\mathbf{R}\left[t\right]⁄\left(t^4+1\right)\mathbf{R}\left[t\right]$$ and $\pi$ the quotient homomorphism $$\pi :\mathbf{R}\to \mathbf{R}\left[t\right]⁄\left(t^4+1\right)\mathbf{R}\left[t\right]\text{;}$$ observe that $A$ is an $\mathbf{R}$-algebra via $\pi$.

   1. Determine the group of $\mathbf{R}$-algebra automorphisms of $A$.

   2. Assuming as known the fact (a consequence of the “fundamental theorem of algebra”) that, up to $\mathbf{R}$-algebra isomorphism, the only non-trivial finite extension of the field $\mathbf{R}$ is $\mathbf{C}$, find all subfields of $A$ that contain $\pi \left(R\right)$.

3. Recall that the multiplicative group of a finite field must be cyclic. For the irreducible polynomial $p\left(t\right)\in F\left[t\right]$ find a polynomial in $F\left[t\right]$ of degree $1$ whose congruence class mod $p\left(t\right)$ determines a generator for the multiplicative group of the finite field $F\left[t\right]⁄\left(p\left(t\right)\right)F\left[t\right]$ when

   1. $F=\mathbf{Z}⁄2\mathbf{Z},p\left(t\right)=t^4+t+1$.

   2. $F=\mathbf{Z}⁄3\mathbf{Z},p\left(t\right)=t^2+1$.

   3. $F=\mathbf{Z}⁄3\mathbf{Z},p\left(t\right)=t^3-t-1$.

   4. $F=\mathbf{Z}⁄2\mathbf{Z},p\left(t\right)=t^5+t^2+1$.

4. Find a monic polynomial $q\left(t\right)$ of degree $4$ with integer coefficients having $$\begin{array}{cccc}& \hfill \alpha & \hfill =\hfill & \sqrt{2}+\sqrt{3}+\sqrt{6}\hfill \\ & \hfill \beta & \hfill =\hfill & -\sqrt{2}+\sqrt{3}-\sqrt{6}\hfill \\ & \hfill \gamma & \hfill =\hfill & \sqrt{2}-\sqrt{3}-\sqrt{6}\hfill \\ & \hfill \delta & \hfill =\hfill & -\sqrt{2}-\sqrt{3}+\sqrt{6}\hfill \end{array}$$ as real roots. Explain why $q\left(t\right)$ must be irreducible in $\mathbf{Q}\left[t\right]$.

5. For each of the following monic polynomials $p$ of degree $4$ with coefficients in $\mathbf{Q}$ determine the extension degree over $\mathbf{Q}$ of the smallest subfield of $\mathbf{C}$ in which all complex roots of $p$ lie:

   1. $t^4-10t^3+35t^2-50t+24$.

   2. $t^4+2$.

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

   4. $t^4+t-1$.

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