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.

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