Posted: November 19th, 2015

Ring Theory

Ring Theory
Assignment 1
1. For each of the following polynomials, determine if it is irreducible or not,
and if it is reducible, factor it into irreducibles.
(i) 2x
4 + 12x
2 – 18x + 15 ? Q[x].
(ii) x
4 + x
3 + 1 ? Z3[x].
2. Find all monic irreducible polynomials of degree 3 in Z3[x].
3. Let R = Z2[x], I = {f ? R | f(c) = 0 for all c ? Z2}. Show that I is an
ideal. Is it a principal ideal? Is it a prime ideal? How many elements does
R/I have?
4. Show that if R is a field, then its field of fractions is isomorphic to R.
5. Let I and J be ideals in a ring R.
(a) Show that I n J is an ideal.
(b) Show that IJ = {
P
i
xiyi
| xi ? I, yi ? J} is an ideal.
(c) Show that I + J = {x + y | x ? I, y ? J} is an ideal.
(d) Let I = h6i and let J = h15i be ideals in Z. For each of the ideals
IJ, I n J and I + J, find a generator of it.
6. Determine all maximal ideals in R = Z2[x]/(x
4 +x
2 +x+ 1), and for each
such maximal ideal I, determine the order of R/I.
7. In each of the following situations, find a greatest common divisor of r
and s and hence find a generator for the ideal (r) + (s).
(a) r = x
3 – 3x
2 + 3x – 2, s = x
2 – 5x + 6 in Q[x].
(b) r = 1 – 5i, s = 1 + 2i in Z[i].
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