Abstract Algebra Answers

a*b=-a-b-2ab; a⊕b=3a+3b

x*y=x²+2x+y²; x⊕y=x+y

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation

defined by aOb = max{ a,b}.

Give an example of a subring of a ring, say A, that is not an ideal of A

Determine which of the polynomials below is (are) irreducible over Q. a. x⁵+9x⁴+12x²+6 b. x⁴+x+1

Show that x³+ x²+x+1 is reducible over Q. Does this fact contradict the corollary to Theorem 17.4?

Determine which of the polynomials below is (are) irreducible over Q. a. x5 1 9x4 1 12x2 1 6 b. x4 1 x 1 1

Let G be a group such that (ab)^p = a^p b^p for all a,b belongs to G, where p is a prime number. Let S= {x belongs to G /x^p^m = e for some m depending on x} . Prove S is a normal subgroup of G

If N is normal in G and a belongs to G is of order O(a), prove that the order, m of Na in G/N is a divisor of O(a)