189-235A: Basic Algebra I Assignment 8

Due: Wednesday, November 20

Section 9.1: 6, 22, 24, 30.

Section 9.2: 12, 13, 15, 32.

Section 9.3: 9, 11.

11. In class, we showed that the ring Z[i] is a Euclidean domain, and hence, a unique factorization domain. We concluded that every prime which is congruent to 1 mod 4 is a sum of two integer squares. In this question, we will prove the following theorem due to Lagrange: Every positive integer can be written as a sum of 4 integer squares. (For example, 2 = 1^{2} + 1^{2} + 0^{2} + 0^{2}, 7 = 2^{2} + 1^{2} + 1^{2} + 1^{2}, etc.)

a) Verify Lagrange’s theorem empirically, by expressing 23 and 107 as a sum of four squares.

b) Recall the ring of Hurwitz integer quaternions:

R = {a + bi + cj + dk},

where a, b, c, d are either all integers or halves of odd integers, and i, j, k satisfy

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

In a previous assignment, you showed that R is a (non-commutative!) ring with identity, and that the “quaternionic norm” δ : R ⟶ {0, 1, 2, …} defined by:

δ(a + bi + cj + dk) = a^{2} + b^{2} + c^{2} + d^{2}

satisfies δ(r_{1}r_{2}) = δ(r_{1})δ(r_{2}) for all r_{1}, r_{2} ∈ R. Using the quaternionic norm, show that if a and b are two integers which are sums of 4 squares, then their product ab is also a sum of four squares. Hence it is enough to prove that every odd prime p is a sum of four squares, to prove Lagrange’s theorem.

From now on, fix an odd prime p.

c) Show that there exists x and y in Z_{p} such that x^{2} + y^{2} + 1 = 0 in Z_{p}. (Hint: note that the expressions of the form x^{2} and −1 − y^{2} each take on exactly (p + 1)/2 distinct values in Z_{p}.) Conclude that there exist integers a and b such that a^{2} + b^{2} + 1 = pm, with m < p.

d) A (non-commutative) integral domain R is said to be Euclidean if there is a function δ : R ⟶ {0, 1, 2, …} such that

1. δ(a) ≤ δ(ab) for all a, b ≠ 0.

2. For all a, b ∈ R, there exist q and r so that

a = qb + r, with δ(r) < δ(b).

This is an extension of the notion of Euclidean domain, to the context of noncommutative rings. Show that the ring R of integral Hurwitz quaternions is Euclidean. (Hint: the proof uses an idea similar to the proof given in class that Z[i] is Euclidean.)

e) A left ideal in a non-commutative ring R is a subset I ⊂ R which is closed under subtraction, and under left multiplication by elements of R: i.e., for all r ∈ R and x ∈ I, rx belongs to I.

A left ideal is said to be principal if it is generated by a single element a: i.e., it consists of all multiples xa of a by some x ∈ R (on the left).

If R is a Euclidean ring (in the sense of d), show that every left ideal in R is principal, by mimicking the proof given in class for commutative rings.

f) Show that p is the quaternionic norm of an element γ of R. (Hint: Let I be the left ideal of R generated by p and by 1 + ai + bj, where a, b are as the integers obtained in part c. I.e., I is the set of elements of the form xp + y(1 + ai + bj), where x and y belong to R. Let γ be a generator of I.)

g) Find all the units in R. Show that the element γ obtained in part f) has an associate whose coefficients are integers. Deduce the Lagrange four squares theorem.