Page 53 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Hence D is closed under multiplication. Notes
(I ) Multiplication is commutative in the set of real numbers.
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(I ) Multiplication is associative in the set of real numbers.
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(I ) 1 0 2 1 D and for a b 2 D , we have
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(1 0 2(a b 2) (a b 2)(I 0 2) a b 2
1 is the multiplicative identity in D.

I . In the set of real numbers multiplication is distributive over addition.
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I . Now, to prove that this ring is without zero divisors let a b 2 and c d 2 be two
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arbitrary elements of D. Then
(a b 2)(c d 2) 0 ac 2bd 0 and bc ad 0
= either a = 0 and b = 0 or c = 0 and d = 0

= either a b 2 or c d 2 = 0.

Thus the given set is a commutative ring with unity and without zero-divisors, i.e., it is an
integral domain.

1.4 Fields

Definition: A commutative ring with unity is called a field if its every non-zero element possesses
a multiplicative inverse.
Thus a ring R in which the elements of R different from 0 form an abelian group under
multiplication is a field.
Hence, a set F, having at least two distinct elements together with two operations ‘+’ and ‘.’ is
said to form a field if the following axioms are satisfied:

(F ) (F, +) is an abelian group.
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(F ) F is closed under addition, i.e., , a b F a b F .
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(F ) Addition is commutative in F i.e., (a b ) c a (b c )
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for all , ,a b c F .
(F ) Identity element with respect to addition exists in F, i.e., , 0 F such that a + 0 = 0 + a = a
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a F.
(F ) There exists inverse of every element of F, i.e., a F, there exists an element –a in F such
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that
a ( a ) ( a ) a 0 .

(F ) Properties of (F,.)
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(F ) F is closed under multiplication, i.e.,  ,a b F , a b F .
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(F ) Multiplication is commutative in F, i.e., a b b a for all a, b, F.
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(F ) Multiplication is associative in F, i.e., (a . b) . c = a . (b . c) for all a, b, c, F.
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