189-245A: Algebra 1

Assignment 4

Due: Monday, November 10.

1. Show that the ring $\mathbb Z$ of integers has no subrings other than itself. (We recall that according to the conventions in this course, a ring contains neutral elements for addition, and for multiplication, and that these two distinguished elements are supposed to be distinct.)

2. Let $n$ be an integer.

(a) Prove that there is a unique homomorphism from $\mathbb Z$ to $\mathbb Z/n\mathbb Z$.

(b) Prove that there is no homomorphism from $\mathbb Z/n\mathbb Z$ to $\mathbb Z$.

(c) What conditions do two integers $n$ and $m$ need to satisfy for there to be a homomorphism from $\mathbb Z/n\mathbb Z$ to $\mathbb Z/m\mathbb Z$?

3. Show that the ideal generated by $3$ and $1+\sqrt{-5}$ is not a principal ideal in the ring $\mathbb Z[\sqrt{-5}]$.

4. Show that the ideal generated by $5$ and $1-8i$ is a principal ideal in the ring $\mathbb Z[i]$ of Gaussian integers, and write down a generator.

5. Show that the ring $\mathbb Z/2\mathbb Z[x]/(x^3+x+1)$ is a field, and find the multiplicative inverse of the element $[x^2+1]$ in this field.

6. Let $F$ be a field, let $R_1 = F[x]$ be the ring of polynomials with coefficients in $F$, and let $R_2$ be the ring of all functions from $F$ to itself, with addition and multiplication defined as the usual operations on functions with values in a ring. The function $$\varphi:R_1\longrightarrow R_2$$ which sends the polynomial $f\in F[x]$ to the $F$-valued function $a\mapsto f(a)$ on $F$ which it induces, is a homomorphism of rings. Describe the kernel of this homomorphism when

(a) $F$ is of infinite cardinality;

(b) $F$ is a finite field of cardinality $q$.

(c) When $F=\mathbb Q$ or $\mathbb R$, show that the homomorphism $\varphi$ is not surjective.

(d) When $F$ is a finite field with $q$ elements, show that the homomorphism $\varphi$ is surjective, by comparing the cardinalities of $R_1/\ker(\varphi)$ and $R_2$ and invoking the isomorphism theorem for rings.

7. Let $n>3$ be an integer. Show that the ring ${\mathbb Q}[x,y,z,\frac{1}{x},\frac{1}{y},\frac{1}{z}]/(x^n+y^n-z^n)$ admits no homomorphism to the rational numbers. (Hint: You may invoke Fermat's Last Theorem proved by Andrew Wiles.)

8. Show that the quaternion group $Q_8 = \{ \pm1, \pm i, \pm j,\pm k\}$ of order $8$ and the dihedral group $D_8 = \{ 1,r,r^2,r^3, D_1, D_2, V, H\}$ (with notations as in class) are not isomorphic to each other.

9. List the conjugacy classes in the groups $Q_8$ and $D_8$ of the previous exercise.

10. What is the maximal order of an element of the permutation groups $S_5$ on $5$ elements? What about the permutation group $S_6$?