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Math Help - element in principal ideal

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by reflection_009 View Post

    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.
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  3. #3
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    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
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  4. #4
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    Quote Originally Posted by reflection_009 View Post

    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 \text{Aut}(\mathbb{Z}/n\mathbb{Z}) \cong (\mathbb{Z}/n\mathbb{Z})^{\times} to prove that the answer is (3). in fact: \text{Aut}(\mathbb{Z}/8\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}.
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  5. #5
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    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?
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  6. #6
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    Quote Originally Posted by reflection_009 View Post

    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 x^3 + x^2 + 1 is irreducible over \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!
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