**Prob.**Show that the polynomial x

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

Solution:Click here.

**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

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

Solution:Click here.

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

Solution:Click here.