Section 6 - Divisibility and Factorization

Question 1

Integral Domain

Answer 1

An integral domain is a commutative ring R such that:
(i) R has an identity 1 != 0
(ii) ∀a,b∈R such that ab = 0, either a = 0 or b = 0

Question 2

Field

Answer 2

A field is a commutative ring R such that:
(i) R has an identity 1 != 0
(ii) ∀a∈R∖{0}, there is b∈R such that ba = 1 (every non-zero element of R is a unit)

Question 3

If n is an integer such that n>=2, what can be concluded about n?

Answer 3

If n >= 2 is an integer, the following are equivalent:
(i) n is prime
(ii) Z/nZ is an integral domain
(iii) Z/nZ is a field

Question 4

Quadratic Ring

Answer 4

Z[√n] = {x + y√n | x,y∈Z}
Z[√n] is a subring of R (real numbers) if n > 0, and a subring of C (complex numbers)

Question 5

Norm Map

Answer 5

N: Z[√n] → Z
x + y√n → x^2 - ny^2

Question 6

What are the 2 key properties of norm maps

Answer 6

(i) N respects multiplications: N(ab) = N(a)N(b) for all a,b∈Z[√n]
(ii) If a∈Z[√n], then a is a unit iff N(a) ∈ {1, -1}

Question 7

Euclidean Function

Answer 7

Let R be an integral domain. A Euclidean function on R is a map Φ: R → Z>=0 such that:
(i) Φ(0R) = 0, and
(ii) ∀a,b∈R with b != 0R, there are q,r∈R such that a = qb + r and either r = 0 or Φ(r) < Φ(b)

Question 8

Euclidean Domain

Answer 8

An integral domain that has a Euclidean function

Question 9

Principal Ideal Domain

Answer 9

An integral domain in which every ideal is principal

Question 10

Prime Element

Answer 10

An element r is called prime if ∀a,b∈R (where R is an integral domain and r is a non-zero element of R that is not a unit) such that r divides ab, either r divides a or r divides b

Question 11

Irreducible Element

Answer 11

An element r is called irreducible if ∀a,b∈R (where R is an integral domain and r is a non-zero element of R that is not a unit) such that r=ab, either a is a unit or b is a unit

Question 12

Integral Domains and Prime Elements Proposition

Answer 12

Let R be an integral domain. Then every prime element of R is irreducible

Question 13

Principal Ideal Domains and Irreducible Elements

Answer 13

Let R be a principal ideal domain. Then, every irreducible element of R is a prime element. Thus, in a principal ideal domain, the notions of prime element and irreducible element are interchangable

Question 14

Rational Root Test

Answer 14

Let f = anxn + · · · + a1x + a0 ∈ Z[x], where n ≥ 1 and an != 0. If r/s is a root of f, where r and s are coprime integers and s != 0, then r | a0 and s | an.

Question 15

Let f be a field, and suppose that f ∈ F[x] has a degree 2 or 3. Then f is irreducible in F[x] iff ________________

Answer 15

Let f be a field, and suppose that f ∈ F[x] has a degree 2 or 3. Then f is irreducible in F[x] iff f has no roots in F

Question 16

Eisenstein’s Criteria

Answer 16

Let f = x^n + (an−1)x^n−1 + · · · + a1x + a0 ∈ Z[x] be a monic polynomial of degree at least 1, and suppose that there is a prime number p such that p | ai for all i ∈ {0, . . . , n − 1} but p^2 ∤ a0. Then f is irreducible in both Z[x] and Q[x].

Question 17

Prime Ideal

Answer 17

Let R be a commutative ring with identity 1 != 0. An ideal P != R is called prime if for all a, b ∈ R such that ab ∈ P, either a ∈ P or b ∈ P

Question 18

Integral Domains and Prime Ideals Proposition

Answer 18

Let R be a commutative ring with identity 1 != 0, and let I be an ideal of R. Then I is prime iff R/I is an integral domain

Question 19

Maximal Ideal

Answer 19

Let R be a commutative ring with identity 1 != 0. An ideal M != R is called maximal if the only ideals of R containing M are M and R.

Question 20

Fields and Maximal Ideals Proposition

Answer 20

Let R be a commutative ring with identity 1 != 0, and let I be an ideal of R. Then I is maximal iff R/I is a field

Question 21

Relationship between Maximal Ideals and Prime Ideals

Answer 21

If R is a commutative ring with identity 1 != 0, then every maximal ideal of R is prime.

Question 22

Associate

Answer 22

Let R be a commutative ring with identity 1 != 0, and let a,b∈R. We say that a is associate to b, written a ~ b, if u is a unit such that a = ub. In that case, b = u^(-1)u, so b~a

Question 23

Unique Factorization Domain

Answer 23

An integral domain R such that for every non-zero element a∈R that is not a unit, the following both hold:
(i) a is a product of irreducible elements (this includes the possibility that a itself is irreducible)
(ii) The factorization of a into irreducibles is essentially unique, in the sense that if π1…πm and π1’…πn’ are two such factorizations, then m=n and, after a reordering of the factors if necessary, πi ~ πi’ for all i∈{1, …, m}

Question 24

Relationship between principal ideal domains and unique factorization domains

Answer 24

Every principal ideal domain is a unique factorization domain

Question 25

Relationship between Euclidean domains and principal ideal domains

Answer 25

Every Euclidean domain is a principal ideal domain

Question 26

Relationship between integral domains and fields

Answer 26

Every field is an integral domain