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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 Infer 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 4dimensional division algebra H over R,
i. e., a
division ring which is a 4dimensional 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 zerodivisors, 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.
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 = 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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