Chapter 1

Question 1

Fundamental Theorem of Arithmetic

Answer 1

Let n≥1 be a natural number. Then the following hold.

(1) n can be written as a product of its primes n = p₁p₂…pₙ
(2) This factorisation is unique

Question 2

Gaussian Integer

Answer 2

A complex number whose real and imaginary parts are both integers
ℤ[i] = {a+bi | a,b∈ℤ}

Question 3

Theorem I.2.1 (factors of primes)

Answer 3

Suppose p is prime and a,b∈ℤ if p|ab then p|a or p|b

Question 4

Corollary of Theorem I.2.1

Answer 4

Suppose p is prime and a₁,a₂,…,aₙ∈ℤ if p|a₁a₂…aₙ then there exists i such that p|aᵢ

Question 5

Euclidean Algorithm on a,b

Answer 5

1. divide a by b. a = cb + r
2. We know hcf(a,b)=hcf(b,r)
3. repeat with b and r until no remainder

Question 6

Bezouts Lemma

Answer 6

if h = hcf(a,b) then there exists x,y∈ℤ s.t. h=ax+yb

Question 7

Division Theorem

Answer 7

Let a and d be natural numbers. Then there are integers q and r such that

a= qd+r

Question 8

does an analogue of the FTA hold for gaussian integers?

Answer 8

yes

Question 9

ring

Answer 9

A set R of complex numbers is a ring if

(i) x+y, x-y, xy ∈R for all x,y∈R
(ii) 0,1∈R

Question 10

a is a factor of b in R (definition)

Answer 10

Let a,b∈R. We say that a is a factor of b in R, a|b, if there exists z∈R s.t. b=az

Question 11

a|b <=>

Answer 11

b/a∈R

b=az

Question 12

does a|0 ?

Answer 12

true

Question 13

if 0|a then

Answer 13

a=0

Question 14

does 1|a

Answer 14

true always

Question 15

Lemma I.4.1 (factors in rings)

Answer 15

Suppose that R is a ring and that a,b,c∈R. If a|b and a|c then a|(b+c).

Question 16

unit

Answer 16

An element u of a ring R is a unit in R if u|1 in R. That is if 1/u ∈ R.

Question 17

what is always a unit and never a unit?

Answer 17

1 is always a unit

0 is never a unit

Question 18

Is it always possible to divide by a unit?

Answer 18

yes. x/u = x * 1/u ∈ R

Question 19

units of ℤ

Answer 19

1 -1

Question 20

units of the rationals

Answer 20

every non-zero element of the rationals is a unit

Question 21

units of ℤ[i]

Answer 21

1,-1,i,-i

Question 22

associate

Answer 22

Let x be an element of a ring R. The associates of x are the elements ux where u is a unit of R.

Question 23

Associate rules:

Answer 23

1. x is always an associate of itself
2. if x is an associate of y then y is an associate of x
3. if x is an associate of y and y is an associate of z then x is an associate of z
   i. e. associate is an equivalence relation

Question 24

equivalence relation

Answer 24

a~a (reflexive)
if a~b then b~a (symmetric)
if a~b and b~c then a~c (transitive)

Question 25

Lemma I.6.1 (factors and associates)

Answer 25

Suppose that a,b,a’,b’ are members of a ring R with a’ an associate of a and b’ an associate of b. Then a|b => a’|b’

Question 26

Lemma I.7.1 (factors of x)

Answer 26

suppose that x is an element of a ring R. Then each unit of R and each associate of x is a factor of x.

Question 27

irreducible

Answer 27

A non-zero non-unit element of a ring R is irreducible in R if the only factors of x in R are the associates of x and the units of R

Question 28

irreducibles in ℤ

Answer 28

the prime numbers and their negatives

Question 29

if x if irreducible then…

Answer 29

all of its associates are irreducible

Question 30

Are 0 and units of R irreducible?

Answer 30

no

Question 31

The norm of α

Answer 31

N(α)=αα*

Question 32

for ℤ[i] N(α) is given by

Answer 32

N(α) = a² + b²

Question 33

N(α)=0 <=> α=0

Answer 33

true

Question 34

Lemma I.8.1 (rules for norms)

Answer 34

Let α,β∈ℤ[i]. Then the following hold.

(i) α|β in ℤ[i] => N(α)|N(β) in ℤ
(ii) if α is a unit <=> N(α)=1
(iii) N(α) is irreducible in ℤ => α is irreducible in ℤ[i]

Question 35

Lemma I.8.2 (units of ℤ[i])

Answer 35

The units of ℤ[i] are 1,-1,i and -i

Question 36

How to determine the factors of n in a ring R=ℤ[i]

Answer 36

1. suppose α|n then there exists β∈ℤ[i] s.t. n=αβ
2. N(n) = N(αβ) = N(α)N(β). since the norms are non-negative integers we have the following (however many factors N(n) has) cases
3. N(α) = 1 => α is a unit
4. N(α) = (factor of n). Write, α= a+bi a,b∈ℤ. Then, a²+b²=n. Since a,b∈ℤ we can extract the possibilities for (a,b). Check which of these are actually factors. by checking n/(a+bi)∈ℤ[i]. If it is a factor then all of its associates are factors.
5. N(α) = (partner factor of n). => N(β) = (factor of n) => β = associate of the factors calculated above => α is an associate of n/β
6. N(α) = N(n). Then N(β) = 1 => β is a unit => α is an associate of n

Question 37

Euclidean Ring

Answer 37

A ring R of complex numbers is a Euclidean Ring (ER) if

(i) |a|²∈ℤ for all a∈R
(ii) given any fraction of the form z=a/d with a,d∈R, d not 0. Then there exists a q∈R s.t. |z-q|²<1
i. e. z has distance less than 1 some some member of R

Question 38

Theorem I.9.1 . example of a ER (1)

Answer 38

The ring ℤ is a ER

Question 39

Theorem I.9.2 example of a ER (2)

Answer 39

The ring ℤ[i] is a ER

Question 40

Lemma I.9.3. units in ER.

Answer 40

Let R be a euclidean ring. Let d∈R and suppose |d|²=1. The d is a unit.

Question 41

Theorem I.9.4. Analogue of FTA in ER.

Answer 41

Let R be a ER and a∈R. We suppose that a is non-zero non-unit. The a is a product of irreducibles.

Question 42

Highest Common Factor in a Ring

Answer 42

Let R be a ring and a,b∈R. A highest common factor is an element h∈R with the properties
(i) h|a, h|b
(ii) for each c∈R if c\a and c\b then c|h
HCF(a,b) denotes the set of all highest common factors

Question 43

Highest Common Factor in a Ring (described)

Answer 43

h is a common factor of a and b where every common factor of a and b is a factor of h. The highest common factor is the set of all these h’s.

Question 44

If h is a common factor then what do we know about other common factors

Answer 44

any associates of h are also common factors

Question 45

Can HCF(a,b) be empty?

Answer 45

yes (except in the integers)

Question 46

How to find HCF(a,b)

Answer 46

1. find all the common factors of a and b
2. for each number:
   - check if its divisible by all the other values: if its not then don’t add to HCF(a,b), if it is do add to HCF(a,b)

Question 47

Theorem I.10.1. The Euclidean Algorithm in a ER

Answer 47

Let R be a Euclidean ring and let a,b be non-zero elements of R. Then:

(i) the numbers a and b have a HCF (i.e. the HCF is non-empty)
(ii) if h is a HCF of a and b then there exists x,y∈R such that h = xa + yb

Question 48

Prime

Answer 48

Let R be a ring. A non-zero non-unit element p∈R is prime if for all a,b∈R we have that p|ab => p|a or p|b

Question 49

Lemma I.11.1. induction of definition of p.

Answer 49

Let p∈R be a prime. If a₁,a₂,…,aₙ∈R and p|a₁a₂…a then there exists an i s.t. p|aᵢ

Question 50

Lemma I.11.2. prime/irreducible.

Answer 50

Let p∈R be prime. Then p is irreducible.

Question 51

Theorem I.11.3

Answer 51

Let p be an element of a Euclidean ring R. Then,

p is irreducible <=> p is prime

Question 52

Unique Factorisation Domain (UFD)

Answer 52

A UFD is a ring R with the property that for each a∈R with a a non-zero non-unit

(i) There is exists irreducibles p₁, … , pₙ s.t. a=p₁…pₙ
(ii) If q₁,…,qₘ are also irreducibles with a = q₁…qₘ then m=n and there is a 1-1 correspondence between the ps and the qs s.t. corresponding numbers are associate.

Question 53

Theorem I.12.1. ERs and UFDs

Answer 53

A Euclidean Ring is a unique factorisation domain

Question 54

Lemma (after ER => UFD)

Answer 54

Let R be a ER and n∈ℕ and suppose that p₁,…,pₙ are irreducibles in R, Suppose that q₁,…,qₘ are irreducibles and u is a unit such that p₁…pₙ=uq₁…qₘ. Then n=m and there is a bijection σ: {1,…,n}->{1,…,m} such that for all 1≤i≤n pᵢ is associate to q_σ₍ᵢ₎