Algebra Review

3 Gaussian integers and quaternions; sums of two squares and
four
squares

Definition 3.1 The Gaussian Integers are complex numbers of the form {a+bi : a, b ∈ Z}
where . They form the ring Z[i]. The norm of z ∈ Z[i] is .

Exercise 3.2 Define divisibility among Gaussian integers. Observe that z |w => N(z) |N(w).
Show that the units among the Gaussian integers are ±1,±i.

Exercise 3.3 Use Gaussian integers to show that sum of two squares .
Hint. Observe that N(zw) = N(z)N(w).

Exercise 3.4 Define division with remainder among Gaussian integers. Show the existence
of g.c.d. ’s. Use this to establish unique prime factorization in Z [i].

Exercise 3.5 Show: if z is a prime in Z[i] then N(z) is either p or p2 for some prime p ∈ Z.
In the former case p = N(z) = a2 + b2; in the latter case, p = z.

Exercise 3.6 Let p ∈ Z be a prime. Prove: p is a prime in Z[i] if and only if p ≡-1
(mod 4). Hint. “If:” if p ≡-1 (mod 4) then p ≠ a2 + b2. “Only if:” if p ≡1 (mod 4) then
. Let w = a + bi ∈ Z[i]. Let z = g.c.d. (p,w).

Exercise 3.7 Infer from the preceding exercise: if p is a prime (in Z) and p ≡1 (mod 4)
then p can be written as a2 + b2.
Exercise 3.8 The positive integer can be written as a sum of two squares if and
only
if .

Exercise 3.9 Show that the number of ways to write n as a2 + b2 in Z is

where ε = 1 if n is a square and 0 otherwise.

Exercise 3.10 Let n be a product of primes ≡ 1 (mod 4) and suppose n is not a square.

Prove: the number of ways to write n as a2 +b2 is d(n) (the number of positive divisors of n).

Definition 3.11 The quaternions form a 4-dimensional division algebra H over R, i. e., a
division ring which is a 4-dimensional vector space over R. The standard basis is denoted by
1, i, j, k, so a quaternion is a formal expression of the form z = a+bi+cj+dk. Multiplication
is performed using distributivity and the following rules :

It is clear that H is a ring. We need to find inverses.

Exercise 3.12 For z = a+bi+cj+dk, we define the norm of z by N(z) = a2 +b2 +c2 +d2.
Prove: , where = a - bi - cj - dk is the conjugate quaternion.

Exercise 3.13
Let z,w ∈ H. Prove: N(zw) = N(z)N(w).

Exercise 3.14

where t, u, v,w are bilinear forms of (a, b, c, d) and (k, l, m, n) with integer coefficients . Calculate
the coefficients .

Exercise* 3.15 (Lagrange) Every integer is a sum of 4 squares. Hint. By the preceding
exercise, it suffices to prove for primes. First prove that for every prime p there exist
such that p and g.c.d. . Let now m > 0 be minimal such
that ; note that m < p. If m ≥ 2, we shall reduce m and thereby obtain
a contradiction (Fermat’s method of infinite descent; Fermat used it to prove that if p ≡ 1
(mod 4) then p is the sum of 2 squares). If m is even, halve m by using and
(after suitable renumbering). If m is odd, take such that .
Observe that and , so where 0 < d < m. Now represent
as a sum of four squares, , using the preceding exercise . Analyzing
the coefficients, verify that . Now , the desired contradiction.

4 Fields

Definition 4.1 A field is a commutative division ring.

Example 4.2 Let F be a field.

:= set of n × n matrices over F is a ring

:= group of units of is called the “General Linear Group

Exercise 4.3 A finite ring with no zero divisors is a division ring. (Hint: use Exercise 2.13.)

Theorem 4.4 (Wedderburn) A finite division ring is a field.

Exercise 4.5 If F is a field and is a finite multiplicative subgroup then G is cyclic.

Definition 4.6 Let R be a ring and for x ∈ R let be the g.c.d. of all n such that nx = 0
where

nx := x + . . . + x when n > 0

nx := −x − . . . − x ( n times ) when

nx := 0 (n times) when n = 0.

Exercise 4.7 · x = 0

Exercise 4.8
If R has no zero divisors then .

Definition 4.9 The common value is called the characteristic of R.

Exercise 4.10 If R has no zero divisors then char(R) = 0 or it is prime. In particular, every
field has 0 or prime characteristic.

Exercise 4.11 If R is a ring without zero-divisors, of characteristic p, then (a+b)p = ap+bp.

Exercise 4.12

1. If R has characteristic 0 then R Z

2. If R has characteristic p then R Z/pZ.

Exercise 4.13 If F is a field of characteristic 0 then F Q.

Definition 4.14 A subfield of a ring is a is a subset which is a field under the same operations.
If K is a subfield of L then we say that L is an extension of K; the pair (K,L) is referred to
as a field extension and for reasons of tradition is denoted L/K.

Definition 4.15 A prime field is a field without a proper subfield.

Exercise 4.16
The prime fields are Q and Z/pZ (p prime).

Definition 4.17 Observe: if L/K is a field extension then L is a vector space over K. The
degree of the extension is [L : M] := dimK L. A finite extension is an extension of finite
degree.

Exercise 4.18
The order of a finite field is a prime power. Hint. Let L be a finite field and
K its prime field, so |K| = p; let [L : M] = k. Prove: |L| = pk.

Exercise 4.19 The degree of the extension C/R is 2. The degree of the extension R/Q is
uncountably infinite (continuum).

Exercise 4.20 Prove that are linearly independent over Q.

Exercise 4.21 If K L M are fields then [M : L][L : K] = [M : K].

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