Page 59 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

1 is multiplicative identity in R. Notes

(F ) Let a b 2 0, i.e., at least one of a and b is non-zero then
22
1 a b 2 a b
= 2 2 2 2
a b 2 (a b 2)(a b 2) a 2b a 2b

= A B 2 , R Where A, B Q

and

a b
A = 2 , B 2
a 2b a 2b

a 2 2b 2 0 as otherwise if a = 0, b = 0 which is impossible due to our assumption for non-zero
element a b 2.
Thus at least one of A and B is non-zero. Hence inverse of a b 2 is a non-zero element
A B 2 in R, because
1
A B 2 a b 2 = (a b 2) 1.
(a b 2)
Thus every non-zero element in R possesses multiplicative inverse.
Hence R is a field.

Example 30: If the operations be addition and multiplication (mod p), prove that the set
{0, 1, 2, ..., p – 1}, (mod p) where p is prime, is a field.
Solution: Let this set be denoted by I (p) which has already be shown a commutative ring with
unity. To prove that I (p) is a field we will have to show that every non-zero element of I (p) is
invertible. Let r I (p) and r 0.
Now r 0 0 (mod p)

r is not divisible by p
r and p are relatively prime.

i.e., there exist integers x, y such that rx + py = 1 implying that
rx 1 (mod p) as py 0 (mod p).

Thus x is inverse of r in I (p).

Hence I (p)is a field.
Self Assessment

13. With addition and multiplication as operation prove that
(i) The set {0, 1} (mod 2) is a field.
(ii) The set {0, 1, 2} (mod 3) is a field.

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