# 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 p^{2} for some
prime p ∈ Z.

In the former case p = N(z) = a^{2} + b^{2}; 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 ≠ a^{2} + b^{2}. “Only if:” if p ≡1 (mod
4) then

. Let w = a + bi ∈ Z[i]. Let z = g.c.d. (p,w).

**Exercise 3.7 I**nfer from the preceding exercise: if p is a prime (in Z) and p ≡1
(mod 4)

then p can be written as a^{2} + b^{2}.

**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 a^{2} + b^{2} 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 a^{2} +b^{2} 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) = a^{2} +b^{2} +c^{2}
+d^{2}.

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}= a

^{p}+b

^{p}.

**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.

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

Exercise 4.16

Exercise 4.16

**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.

**The order of a finite field is a prime power. Hint. Let L be a finite field and**

Exercise 4.18

Exercise 4.18

K its prime field, so |K| = p; let [L : M] = k. Prove: |L| = p

^{k}.

**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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