A ring R with at least two elements is called a field if,

1)      It is commutative

2)      It has unity

3)      Each non-zero element possesses multiplicative inverse.

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Problems based on above study:

Prob. Prove that a ring R is commutative, if and only if

( a + b )2 = a2 + 2ab + b2 ,  a, b    R.

Prob. Show that the polynomial x2 + x + 4 is irreducible over F, the field of integer modulo 11.

Prob. If R is a ring, such that a2 = a,  a   R. Prove that,
1)      a + a = 0,  a   R
2)      a + b = 0 implies a = b  a, b   R
3)      R is a commutative ring.

Prob. Define field and prove that the set F = { 0, 1, 2, ….,6 } under addition and multiplication modulo 7 is a field.

Prob. Define field. Prove that the set {0, 1, 2} (mod 3) is a field with respect to addition and multiplication (mod 3).


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