Abstract Algebra Answers

Show that if :f Q->Q is a ring homomorphism, then f(x) = x ∀x ∈ Q. Would

this still be true if f were a group homomorphism? Why, or why not?

Show that Z[√-2] is not a UFD.

Z ring of integers, R ring, f: ring homomorphism from Z to R which is injective. Also if n is not in 13Z, then f(n) is unit in R. Can R be integral domain. One example of such a map.

Prove or disprove that 2 + (11)^1/2∈Q .

Check whether the sequence, {Sn} where


Sn=1/1!+1/3!+1/5!+....1/(2n+1)!

Is convergent or not.

Show that the roots of the equation

Z^4-1=0

form a cyclic group of order 4

Given that x*y=1+x+y, show wheather the binary operation is associative algebrically

Prove or disprove:

a) every left simple ring is right simple

b) every simple ring is left simple

Find all primes p for which −5 is a quadratic residue.