![IRREDUCIBLE POLYNOMIAL
We know that F[x] is an integral domain with unity and
contains F as a proper sub ring. A polynomial f(x) in F[x] is
called irreducible if the degree of f(x) ≥ 1 and, whenever f(x)
= g(x)h(x),where g(x),h(x) € F[x] , then g(x) € F or h(x) € F . If
a polynomial is not irreducible, it is not irreducible, it is called
reducible.
Properties of F[x]:-
1. The division algorithm holds in F[x]. This means that if
f(x) € F[x] and 0 ≠ g(x) € F[x] , then there exist unique
q(x) , r(x) € F[x] such that f(x) = g(x)q(x) + r(x), where
r(x)=0 or degree r(x) < degree g(x).
2. F[x] is a Principal ideal domain
3. F[x] is Unique factorization domain
4. The unit of F[x] are the non zero element of F.](https://image.slidesharecdn.com/m-200503021706/85/Abstract-algebra-Algebraic-closed-field-unit-2-M-Sc-l-semester-Maths-4-320.jpg)
![FIELD
(F,+,.) is called field if
(1)(F,+) Is an abelian group
(2)(F,.) Is an abelian group
(3)F has the distributive properties
i.e. a.(b + c) = a . b + b . c [left distributive
law]
And
(a + b).c = a . c + b . C [right distributive law]
Sub field
If (F,+,.) is a field and K c F (K,+,.) is called
sub field of F. If
1. a , b ∈ K => a-b ∈ K](https://image.slidesharecdn.com/m-200503021706/85/Abstract-algebra-Algebraic-closed-field-unit-2-M-Sc-l-semester-Maths-5-320.jpg)
![FINITE EXTENSION FIELD
If [E:F] = n ,then E is called finite extension of F
OR
If [K:F] = m ,then K is called finite extension of E
Note:-
If [k:F] = m , then (α1 , α2 , -----, αm ) is a basis of K over F.
ALGEABRIC ELEMENT
An element α is said to be algebraic element if it
satisfies the non zero polynomial;
i.e. if α is a root of the polynomial a0 + a1x + a2x2
+----+anxn then α is called algebraic element .](https://image.slidesharecdn.com/m-200503021706/85/Abstract-algebra-Algebraic-closed-field-unit-2-M-Sc-l-semester-Maths-6-320.jpg)
![THEOREM
STATEMENT:-
For any field K the following are
equivalent:
(1)K is algebraically closed
(2)Every irreducible polynomial in K[x] is of
degree 1.
(3)Every polynomial in K[x] factors
completely into linear factors.
(4)Every polynomial in K[x] of positive
degree has at least one root in K.](https://image.slidesharecdn.com/m-200503021706/85/Abstract-algebra-Algebraic-closed-field-unit-2-M-Sc-l-semester-Maths-10-320.jpg)
Abstract algebra - Algebraic closed field, unit - 2 , M.Sc. l semester Maths
- 1. By - Mrs. PREETI SHRIVASTAVA PRESENTATION ON A ALGEABRICALLY CLOSED FIELD “ADVANCED ABSTRACT ALGEBRA” M.Sc. MATHEMATICS – 1ST SEM
- 2. CONTENTS INTRODUCTION IRREDUCIBLE POLYNOMIAL FIELD, SUB FIELD FINITE EXTENSION FIELD, ALGEBRAIC ELEMENT ALGEBRAIC EXTENSION FIELD ALGEBRAICALLY CLOSED FIELD EXAMPLE AND THEOREM
- 3. INTRODUCTION Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of Associative, Commutative and Distributi ve hold. The concept of fields was first (implicitly) used to prove that there is no general formula expressing in terms of radicals the roots of a polynomial with rational coefficients of degree 5 or higher.
- 4. IRREDUCIBLE POLYNOMIAL We know that F[x] is an integral domain with unity and contains F as a proper sub ring. A polynomial f(x) in F[x] is called irreducible if the degree of f(x) ≥ 1 and, whenever f(x) = g(x)h(x),where g(x),h(x) € F[x] , then g(x) € F or h(x) € F . If a polynomial is not irreducible, it is not irreducible, it is called reducible. Properties of F[x]:- 1. The division algorithm holds in F[x]. This means that if f(x) € F[x] and 0 ≠ g(x) € F[x] , then there exist unique q(x) , r(x) € F[x] such that f(x) = g(x)q(x) + r(x), where r(x)=0 or degree r(x) < degree g(x). 2. F[x] is a Principal ideal domain 3. F[x] is Unique factorization domain 4. The unit of F[x] are the non zero element of F.
- 5. FIELD (F,+,.) is called field if (1)(F,+) Is an abelian group (2)(F,.) Is an abelian group (3)F has the distributive properties i.e. a.(b + c) = a . b + b . c [left distributive law] And (a + b).c = a . c + b . C [right distributive law] Sub field If (F,+,.) is a field and K c F (K,+,.) is called sub field of F. If 1. a , b ∈ K => a-b ∈ K
- 6. FINITE EXTENSION FIELD If [E:F] = n ,then E is called finite extension of F OR If [K:F] = m ,then K is called finite extension of E Note:- If [k:F] = m , then (α1 , α2 , -----, αm ) is a basis of K over F. ALGEABRIC ELEMENT An element α is said to be algebraic element if it satisfies the non zero polynomial; i.e. if α is a root of the polynomial a0 + a1x + a2x2 +----+anxn then α is called algebraic element .
- 7. ALGEBRAIC EXTENSION FIELD An extension E is said to be algebraic extension of F if every element of E is algebraic over F. ALGEBRAICALLY CLOSED FIELD A field is said to be algebraically closed if it posses no proper algebraic extension OR A field K is said to be algebraically closed if every algebraic extension of K is itself K.
- 8. EXAMPLE As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in
- 9. Also, no finite field F is algebraically closed, because if a1, a2, …, an are the elements of F, then the polynomial (x − a1)(x − a2) ··· (x − an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.
- 10. THEOREM STATEMENT:- For any field K the following are equivalent: (1)K is algebraically closed (2)Every irreducible polynomial in K[x] is of degree 1. (3)Every polynomial in K[x] factors completely into linear factors. (4)Every polynomial in K[x] of positive degree has at least one root in K.
- 11. THANK YOU