Chapter 1 Definitions

Question 1

Divides

Answer 1

a|b ⇔ there exists a c ∈ Z s.t b = a c .

Question 2

(a) multiples

Answer 2

(a) = aZ = { ka | k∈ Z}

Question 3

An Ideal

Answer 3

An ideal in a commutative ring R is a subset I of R satisfying:

(i) 0 ∈ I
(ii) a, b∈ I ⇒ a±b ∈ I
(iii) a ∈ I , r ∈ R ⇒ ra ∈ I

Notation: I *triangle* R

Question 4

Principle ideal

Answer 4

The set of all multiples of a fixed element a of R is the principle ideal (a) or aR.

a generates (a).

Question 5

Associates of a

Answer 5

Associates of a are the elements,

b = u a

where u is a unit of R.

[Also generators of (a)]

Question 6

A Principle Ideal Domain

Answer 6

A PID is a non-zero, commutative ring such that:

(i) ab=0 ⇒ a=0 or b=0,
(ii) Every ideal in R is principle.

Question 7

The greatest common divisor (gcd)

Answer 7

The gcd of a,b∈Z is the integer d≥0 satisfying:

(i) d|a and d|b
(ii) c|a and c|b ⇒ c|d
(iii) d=au+bv where u,v ∈ Z (Besout’s identity)

Question 8

a,b Coprime

Answer 8

a,b are coprime if gcd(a,b)=1

Question 9

Ideal of a finite sequence

Answer 9

Ideal of a finite sequence

I=(a₁, a₂, … ,a_n) = { k₁a₁, k₂a₂, … ,k_na_n | K_i ϵ Z }

is:

I=(d) where d = gcd(a₁, a₂, … ,a_n).

Question 10

(a₁, a₂, …,a_n) are pairwise coprime

Answer 10

(a₁, a₂, …,a_n) are pairwise coprime if gcd(a_i, a_j) = 1 for i≠j.

Question 11

Prime and composite numbers

Answer 11

A prime number is a an integer P>1 whose only divisors are ±1 and ±P.

Integers which are not prime are called composite.

Question 12

ord_p(n)

Answer 12

ord_p(n) = e ⇔ p^e | n and p^e+1 -|- n

Question 13

An Integral Domain ID

Answer 13

A non-zero, commutative, ring, R, is an Integral Domain if for a,b in Z, ab=0 ⇔ a=0 or b=0

Question 14

An ED

Answer 14

A non-zero ring R is a Euclidean Domain ED if it is an integral domain equipped with a function V: R{0} → |N₀ s.t for a,b in R, with a≠0, there exists q, r in R s.t b=aq+r with ether r=0 or V(r) < V(a)

Question 15

Gaussian Integers

Answer 15

Z[i] = {a+bi | a,b in Z}

Question 16

Euclidean Function Norm

Answer 16

N(a) = aa¯ = c² +d²

^{a= c+id}

Question 17

gcd in a ring

Answer 17

In a ring R, a gcd of two elements a, b is an element d satisfying:

(i) d|a and d|b
(ii) c|a and c|b ⇒ c|d

Question 18

Irreducible

Answer 18

In an Integral Domain, an element p is called irreducible if it is neither 0 nor a unit and p=ab ⇒ either a or b is a unit

Question 19

Prime (In an integral domain)

Answer 19

In an integral domain R, p is called prime if it is neither 0 nor a unit and p|ab ⇒ either p|a or p|b.

Question 20

A Unique Factorisation Domain UFD

Answer 20

An ID, R, is a UFD if:

1. Every non zero element may be expressed as a unit times a product of irreducibles.
2. The factorisation in (1) is unique up to factors and replacing the irreducibles by associates.
   That is, if a in R is nonzero and
   a = u p₁p₂p₃….p_r = u q₁q₂….q_s
   with units u,v and all p_i, q_i irreducible then r=s and after permuting the q_i as necessary, there are v_j units, 1 <= j <= r such that q_j=v_jp_j and u = v v₁v₂…v_r.