**FIELD**

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.

**Problems based on above study:**

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

( a + b )

^{2 }= a^{2}+ 2ab + b^{2}, ∀ a, b ∈ R.**Prob.**Show that the polynomial x

^{2}+ x + 4 is irreducible over F, the field of integer modulo 11.

**Prob.**If R is a ring, such that a

^{2}= 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).

**SEE ALSO:**