# Thread: element in principal ideal

1. ## element in principal ideal

the number of elements of a PRINCIPAL IDEAL DOMAIN can be
(1)15
(2)25
(3)35
(4)36
please give reason why?

2)which of ideal of the ring Z[i] of gaussian integer os Not maximal?
1)<i+i>
2)<1-i>
3)<2+i>
4)<3+i>

3)if Z(G) denote the center of a group G,then order of quotient group G/G(Z) cannot be
1) 4
2) 6
3) 15
4) 25

please give the reasons of above questions?i'm totally confused

2. Originally Posted by reflection_009

the number of elements of a PRINCIPAL IDEAL DOMAIN can be
(1)15
(2)25
(3)35
(4)36
please give reason why?
any finite integral domain is a field and the order of a finite field is a power of a prime. so (2) is the answer.

2)which of ideal of the ring Z[i] of gaussian integer is Not maximal?

1)<i+i> ?
2)<1-i>
3)<2+i>
4)<3+i>
<3 + i> is not maximal because it's not prime! (1 + i)(2 - i) is in <3 + i>, but neither 1 + i nor 2 - i are in <3 + i>. so (4) is the answer.

3)if Z(G) denote the center of a group G,then order of quotient group G/G(Z) cannot be
1) 4
2) 6
3) 15
4) 25
every group of order 15 cyclic. we know that if G/Z(G) is cyclic, then G will be abelian, i.e. |G/Z(G)| = 1. so (3) is the answer.

3. ## automorphism

let Aut(G) denote group of automorphism of a group G.which one of the fillowing is NOT a cyclic group?
1)Aut(Z4)
2)Aut(Z6)
3)Aut(Z8)
4)Aut(Z10)
please give reason

4. Originally Posted by reflection_009

let Aut(G) denote group of automorphism of a group G.which one of the fillowing is NOT a cyclic group?
1)Aut(Z4)
2)Aut(Z6)
3)Aut(Z8)
4)Aut(Z10)
please give reason
use the isomporphism $\displaystyle \text{Aut}(\mathbb{Z}/n\mathbb{Z}) \cong (\mathbb{Z}/n\mathbb{Z})^{\times}$ to prove that the answer is (3). in fact: $\displaystyle \text{Aut}(\mathbb{Z}/8\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}.$

5. thanks ALOT

ONE MORE PROB PLEASE SOLVE IT
Z2[X]/<X^3+X^2+1> Is field or not if yes then how many elements in it
eight,nine or infinite pls help?

6. Originally Posted by reflection_009

thanks ALOT

ONE MORE PROB PLEASE SOLVE IT
Z2[X]/<X^3+X^2+1> Is field or not if yes then how many elements in it eight,nine or infinite pls help?
the polynomial $\displaystyle x^3 + x^2 + 1$ is irreducible over $\displaystyle \mathbb{Z}_2.$ so your quotient ring is a field and it has 8 elements. i'm strating to feel a little bit "uncomfortable" with answering these "multiple choice"

questions especially because you really don't seem to be interested in knowing the whys!