Algebra

Domains and Extension Field
Ashaq Ali
P.G Department of Mathematics

Maulana Azad Memorial College Jammu
presentation, November 28, 2023
Ashaq Ali (CLUJ) Domains and Extension Field External Viva 2023 1 / 17
Table of Contents
1 Domains
2 Polynomial over a Ring
3 Extension of a field
Definition
Let R be an integral domain (I.D) with unity element 1 then an element
u ∈ R is called a unit if ∃ an element v ∈ R s.t u.v = 1.
.Every unit is not unity but every unity is a unit. In Z; 1 − 1 are the only
units.
Definition
Two non-zero elements a and b in integral domain with unit element, are
said to be associates of each other if b = au for some unit u in R
1. In Z, a and -a are associate to each other because a = (−a)(−1) where
-1 is a unit in Z
2. In Z[ι] 1 − ι, 1 + ι are associate to each other because 1 + ι = (1 − ι)ι
Theorem
let R be an I.D with unity ,if d1 = g .c.d(a, b) in R Then d2 is also a g.c.d
(a,b) iff d1 and d2 are associates.
Definition
Let R be an I.D with unit element. A non zero element a in R is said to be
Irreducible if a is not a unit in R and the only divisor of a are units and
associates of a.
Definition
let R be a commutative ring with unity an element p ∈ R is called a prime
element if
1. p ̸= 0, p is not a unit, and
2. if p|ab then either p|a or p|b ∀ a, b ∈ R
Definition
Let R be an I.D with unit element and let a ̸= 0 ∈ R then a has always
divisor namely associates of a and units .These divisor of a are called
Improper divisor.
Definition
A divisor d of a is called a proper divisor if d is not a unit in R and d is not
an associates of a
Definition
Let R be an I.D with unit element then a,b ∈ R are said to be relatively
prime p then g.c.d is unit .
Euclidean Domain
Definition
Let R be an I.D then R is said to be E.D If ∃ a function
d : R − {0} → Z+
Satisfying the following properties (i) d(a) ≥ 0 ∀ a ∈ R − 0 (ii)
d(ab) ≥ d(a) ∀ a, b ∈ R − 0 (iii) Given a,b ∈ R, a ̸= 0∃q, r ∈ R such
that
a = bq + r
where r = 0 or d(r ) < d(b)
example: Z is an E.D. Principal Ideal Domain
Definition
An integral domain R is said to be a Principal Ideal Domain if every
ideal is a Principal ideal i.e I = Ra for some a ∈ R.
Example
1. The set Z, C, R, Q is a P.I.D
2. The Z[ι] is a P.I.D
Theorem
Prove that every Euclidean Domain is a Principal Ideal Domain.
Factorization Domain
Definition
Factorization Domain(F.D): An integral domain R is said to be
factorization domain if every non - unit ’a’ in R can be expressed as a
product of Irreducible elements.
Every E.D and P.I.D is a factorization domain.
Definition
Unique Factorization Domain (U.F.D): An I.D is said to be U.F.D if every
non- unit a ∈ R as a product of Irreducible element
or
An I.D R is said to be U.F.D If
1. Every non - unit a ∈ R can be expressed as a product of Irreducible
elements.
2. whenever a = p1 .p2 ........pn and a = q1 .q2 ......qm are two decomposition
of a in terms of two sets of irreducible then n = m and pi′ s and qi′ s are
asssociate of each other.
Example
The ring < Z, +, . > of integers is a U.F.D
Example
Give an example√of an I.D which√ is not U.F.D.
Sol: let R = Z[ −5] = {a + b −5; a, b ∈ Z}.
This R is an I.D but not a U.F.D because 9 = 3.3
√ √
9 = (2 + −5)(2 − −5)
are two distinct factorization of 9 in terms of Irreducible elements

√
Every P.I.D is not E.D. Counter example: R = Z[12(1 + −19)] is an
example of a P.I.D which is not a E.D
Every U.F.D is not P.I.D Counter example: The ring K[X , Y ] of
polynomial in two variables is a U.F.D but not a P.I.D.
Definition
: A polynomial over a ring R is defined as an infinite ordered system
(a0 , a1 , a2 , ......ai , ....) with almost a finite no. of non zero elements.
A polynomial (a0 , a1 , a2 ......ai ....) is usually denoted by f(x) and can be
written as
f(x) = a0 + a1 x + a22 + .....ai x i
Here f(x) is called a polynomial in one - indeterminate over R.
a0 , a1 x, a2 x 2 ..... are called as terms of the polynomial and a0 , a1 , a2 ..... are
called coefficient of the terms.
Definition
Unit: A unit of F[X ] is an element of f(x)∈ F[X ] which has a
multilpicative inverse ,where F[X ] is a set of all polynomial over the field F
Definition
Associate: if f(x)/g(x) and g(x)/f(x) then f(x) and g(x) are associate
Definition
Divisor element of F[X ]: f(x) is said to be divisor or factor of g(x) if ∃ a
polynomial say h(x)∈ F[X ] S.t
g(x) = f(x).h(x)
Definition
Proper and Improper divisor : let f (x) ∈ F[X ] is divisible by its associate
and units and these divisor are called improper divisor .All the other divisor
are called as proper divisor.
Definition
Prime or Irreducible polynomial: An element √ f(x)∈ F[X
√ ] is prime over F if
3
it has no proper divisor e.g x − 3 = (x + 3)(x − 3) is
prime/irreducible over the field of Rational
where as x 2 − 3 is reducible over a field of real no. (x 2 + 3) over the field
of real irreducible √ √
∴ x 2 + 3 = (x + ι 3)(x − ι 3
Definition
Divisor element of F[X ]: f(x) is said to be divisor or factor of g(x) if ∃ a
polynomial say h(x)∈ F[X ] S.t
g(x) = f(x).h(x)
Definition
Proper and Improper divisor : let f (x) ∈ F[X ] is divisible by its associate
and units and these divisor are called improper divisor .All the other divisor
are called as proper divisor.
Definition
Prime or Irreducible polynomial: An element √ f(x)∈ F[X
√ ] is prime over F if
3
it has no proper divisor e.g x − 3 = (x + 3)(x − 3) is
prime/irreducible over the field of Rational
where as x 2 − 3 is reducible over a field of real no. (x 2 + 3) over the field
of real irreducible √ √
∴ x 2 + 3 = (x + ι 3)(x − ι 3
Theorem
If R is a commutative ring with unity then R[X ] is also a commutative
ring with unity
Theorem
If R is an I.D then R[X ] is also an I.D.
Theorem
If F is a field then F[X ] is an I.D
Theorem
if F is a field then F[X ] is nor a field
Theorem
If R is an I.D with unity ,then units of RandR[X ] are same
Definition
let R be an integral domain with unity . A polynomial f(x)∈ R[X ] of
postive degree (i.e, of deg≥1) is said to be an irreducible polynomial over
R if it can be expressed as a product of two polynomial of postive degree.
In other wordes , if whenever f(x) = g(x)h(x) then deg g = 0 or deg h = 0
A polynomial of postive degree which is not irreducible is called reducible
over R
Theorem
R is a U.F.D⇒ R[x] is a U.F.D
Theorem
Every irreducible element in R[X ] is an irreducible polynomial,R being an
integral domain with unity.
Theorem
If R is a U.F.D then any f(x)∈ R[X ] is an irreducible element of R[X ] iff
either f is an irreducible element of R or f is an irreducible primitive
polynomial of R[X ]
Theorem
Every irreducible element in R[X ] is an irreducible polynomial,R being an
integral domain with unity.
Theorem
If f (x) ∈ R[x] is both primitive and irreducible element of R[x] then f(x) is
irreducible element of K[x].
Definition
Extension of a Field. Let K and F be any two fields and σ : F → K be
monomorphism then F ∼ = σ(F ) ⊆ K . Then (K , σ) is an extension of a field
F. Since F ∼= σ(F ) and σ(F ) is a subfield of k , so we may regard F as a
subfield of K. So, if K and F are two fields such that F is a subfield of K
then K is called an extension of F and we denote it by K |F
i) Every field is an extension of itself.

(ii) Every field is an extension of its every subfield, for example, R is a field
extension of Q and C is a field extension of R.
Definition
Transcendental Number. A number (real or complex) is said to be
transcendental if it does not satisfy any polynomial over rationals, for
example, π,e . Note that every transcendental number
√ is an irrational
number but converse is not true. For example, 2 is an irrational number
but it is not transcendental because it satisfies the polynomial x 2 − 2
Definition
Algebraic Number. Let K |F be any extension. If a ∈ K and a satisfies a
polynomial over F, that is, f (a) = 0 where
f (x) = λo + λ1 x + λ2 x 2 + ..... + λn x n ; λi ∈ F .Then, a is called algebraic
over F. If a does not satisfy any polynomial over F, then a is called
transcendental over F. For example, π is transcendental over set of
rationals but π is not transcendental over set of reals.
very element of F is always algebraic over F.
Example
R|Q is an infinite extension of Q, or [R : Q] = ∞.
Definition
Algebraic Extension. The extension K |F is called algebraic extension if
every element of K is algebraic over F. otherwise, K |F is said to be
transcendental extension if atleast one element is not algebraic over F.
Theorem
Every finite extension is an algebraic extension.
Theorem
:Eisenstein Criterion of Irreducibility. Let
f (x) = ao + a1 x + a2 x 2 + ...... + an x n where ai ∈ Z, an ̸= 0 . Let p be a
prime number such that p|ao , p|a1 , ......, p ∤ an and p 2 ∤ ao , then f(x) is
irreducible over the rationals.
Theorem
:Factor Theorem. LetK |F be any extension and f (x) ∈ F [X ], then the
element a ∈ K is a root of polynomial f(x) iff (x − a)|f (x) in K [x], that is,
iff there exists some g (x) in K [x] such that f (x) = (x − a)g (x).

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Domains and Extension Field

P.G Department of Mathematics

presentation, November 28, 2023

2 Polynomial over a Ring

are two distinct factorization of 9 in terms of Irreducible elements

i) Every field is an extension of itself.

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