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Prove that all real of the form a+b2^1/2,a,b belongs to Z forms a ring

Let G be a group of order 11 2 :1 32 . H ow m any 11 -sylow subgroups and 13 sylow subgroups are there in G ?

If G is the abelian group of integers in the mapping T: G → G given by T(x ) = x then prove that as an automorphism

real of the form a + b 2 , a; b ∈ Z form s a ring.

Let f(x)=x3 +x2 +x+1 be an element of Z [x] . Write f(x) as a product of 2

irreducible polynomials over Z2.

Let f(x) = x3 + 6 be an element Z [x]. Write f(x) as a product of irreducible 7

polynomials over Z7.

Show that the polynomial 2x + 1 in Z4[x] has a multiplicative inverse in Z4[x].

9.

A) Let U(10)={1,3,7,9} be a group under multiplication modulo 10, what is the order of group?

B) What is the order of group Z of integers under addition?

7. For each operation * given below, determine whether * is a binary operation, commutative or associative. In the event that * is not a binary operation, give justification for this.

i. On Z, a*b=a-b

ii. On Q, a*b=ab+1

iii. On Q, a*b=ab/2

iv. On Z+ , a*b= 2ab

Prove that the set Z of all integers with binary operation * defined by a*b=a+b+1

for all a, b belonging to G is an Abelian group.

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