189-235A: Algebra I Assignment 4

Due: Monday, November 4

1. Let F be a (commutative) field. Show that the set R of matrices of the

form ( \begin{matrix}a & b \\ 0 & c \end{matrix} ), with a, b, c ∈ F, is a subring of the ring M_{2}(F) of 2 × 2

matrices with entries in F. Write down a non-trivial two-sided ideal I of R. Describe, in as concrete and explicit a way as you possibly can, the quotient ring R/I. (Your description should be concrete and explicit enough, for example, to make the fact that R/I is a commutative ring completely transparent.)

2. Show that the ring 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.)

3. Let n be an integer.

(a) Prove that there is a unique homomorphism from Z to Z/nZ.

(b) Prove that there is no homomorphism from Z/nZ to Z.

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

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

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

6. Show that the ring Z/2Z[x]/(x^{4} + x + 1) is a field.

7. Find the multiplicative inverse of the element [x^{2}+1] in the field Z/2Z[x]/(x^{4}+ x + 1) of the previous exercise.

8. 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. Show that the function

φ : R_{1} ⟶ R_{2}

which sends the polynomial f ∈ F [x] to the F -valued function a ↦ f(a) on F which it induces, is a homomorphism of rings.

9. When F = Q or R, show that the homomorphism φ of exercise 8 is injective but not surjective^{1}.

10. When F = Z/pZ is the finite field with p elements, show that the homomorphism φ of exercise 8 is not injective. What is its kernel? Show that φ is surjective in this case.

1The injectivity of φ holds for any field F of infinite cardinality, and your proof should give this more general statement without too many efforts. The failure of surjectivity is also true for any infinite field F, but is a bit more tricky to establish. You may like to try your hand at doing this!