305 -Homework set 7

This set is due April 10 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Let $R$ be a commutative ring with identity. Let $I$ be an ideal of $R,$ and let $i$ be a unit of $R.$ Show that $I=R$ iff $i\in I.$ Conclude that the only ideals of a field ${\mathbb F}$ are $\{0\}$ and ${\mathbb F}.$ Also conclude that ${\mathbb F}[x]/I\cong\{0\}$ if $I$ is the ideal of ${\mathbb F}[x]$ generated by a constant nonzero polynomial.

2. Suppose $p(x)\in{\mathbb F}[x]$ is a nonconstant, not irreducible polynomial, and let $I=(p)$ be the ideal generated by $p.$ Show that ${\mathbb F}[x]/I$ has zero divisors.

3. Find an irreducible polynomial $p(x)$ in ${\mathbb Z}_2[x]$ of degree 3, and explicitly show that ${\mathbb Z}_2[x]/(p)$ coincides with (i.e., is isomorphic to) the field of 8 elements built in a previous homework set.

4. Either build a field of 9 elements using the kinds of arguments in the previous problems; or determine a subfield of ${\mathbb C}$ isomorphic to ${\mathbb Q}[x]/(x^3+3x+3).$ (You may assume that $x^3+3x+3$ is irreducible in ${\mathbb Q}[x].$)

4 Responses to 305 -Homework set 7

1. […] Homework 7, due April 10, at the beginning of lecture.  […]

2. Tommy says:

Hi, I finished problem 1, and looking back I just want to make sure the assumptions I make are valid. For all three parts of 1, am I allowed to assume that the unit i is in I? I assume I am, but I don’t want to lose points for something so silly. Thanks

3. Tommy says:

I am a bit confused concerning problem 4. Is it even possible to build such a field? I know I can construct a field of 27 elements, but to construct a field of 9 elements with a characteristic of 3, wouldn’t the polynomial have to be quadratic? Thanks again!

4. Hi Tommy,

I’m not sure I understand your question on problem 1. Email me, and tell me in some detail what you are doing, just to be sure.

About problem 4, you are correct: If we are to build a field of 9 elements, then the polynomial in ${\mathbb Z}_3[x]$ we need to consider must be quadratic and irreducible.