189-235A: Basic Algebra I Assignment 5

Due: Wednesday, October 13

1. Perform the division algorithm for dividing f(x) = 3x^{4} − 2x^{3} + 6x^{2} − x + 2 by g(x) = x^{2} + x + 1 in Q[x]. (I.e., find polynomials q(x) and r(x) with deg(r) < deg(g) satisfying f = gq + r.

2. Same question as 1, with f(x) = x^{5} − x + 1 and g(x) = x^{2} + x + 1 in Z_{2}[x].

3. Let f : Z[x] → Z be the function which to any polynomial p(x) = a_{0} + a_{1}x + ⋯ + a_{d}x^{d} associates its leading term a_{0}: f(p) = a_{0}. Show that f is a homomorphism of rings.

4. Find the gcd of x^{4} − x^{3} − x^{2} + 1 and x^{3} − 1 in Q[x] using the Euclidean algorithm.

5. Find the gcd of x^{4} + 3x^{3} − 2x + 4 and x^{2} + 1 in Z_{5}[x] using the Euclidean algorithm.

6. Let C denote the field of complex numbers, and let \overline{z} = a − bi denote as usual the complex conjugate of the complex number z = a + bi. Let H be the subset of M_{2}(C) consisting of matrices of the form

( \begin{matrix}z_{1} & z_{2} \\ −\overline{z_{1}} & \overline{z_{2}} \end{matrix} ) ,

where z_{1} and z_{2} are complex numbers. Show that H is a subring of M_{2}(C).

7. Show that the ring H of exercise 6 is a non-commutative ring in which every non-zero element has a multiplicative inverse. (In other words, H is a non-commutative field.) The ring H is called the field of quaternions.

The next few questions are optional.

8. Let H′ be the set of all expressions of the form a+bi+cj +dk, where a, b, c and d are real numbers, and i, j, k are formal variables. Define a multiplication on H′ by combining the usual rules for addition and multiplication of real numbers with the rules

i^{2} = j^{2} = k^{2} = −1, ij = −ji = k, ki = −ik = j, jk = −kj = i.

Show that H′ is isomorphic to the ring H of exercise 7. (So from now on, we will write H instead of H′.)

9. Let H be the ring of real quaternions introduced in the previous exercise. A quaternion is said to be integral if it is of the form a1 + bi + cj + dk, where a, b, c, d are integers. Let R be the set of integral quaternions. Show that R is a subring of H.

10. Define a “complex conjugation” on H by the rule

\overline{a1 + bi + cj + dk} := a1 − bi − cj − dk.

Show that if α and β are two quaternions, then

\overline{αβ} = \overline{β} · \overline{α}.

(Note the change in the order of multiplication!)

11. If α belongs to H, define ||α|| := α\overline{α}. Show that if α = a1 + bi + cj + dk, then

||α|| = a^{2} + b^{2} + c^{2} + d^{2}.

Note in particular that if α belongs to the ring R of integral quaternions, then ||α|| ∈ Z.

12. Show that ||αβ|| = ||α|| · ||β||. (Remember in your proof that multiplication in H is not commutative!)

13. Using 12, show that, if m and n are integers which can be expressed as a sum of 4 integer squares, then their product mn can also be expressed as a sum of four integer squares. Use your proof to express 161 = 7 · 23 as a sum of four squares.

This last exercise illustrates the usefulness of the ring R for number theory, and in particular for the study of representations of integers as sums of four squares. A deeper study of the ring theoretic structure of R leads to the following beautiful theorem of Lagrange: Every positive integer can be expressed as a sum of four squares. You should try to test this theorem empirically to get a feeling for what it says. Try also to find a proof!