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Math Help - field ideal question

  1. #1
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    field ideal question

    Let I denote the field generated by x^4+x^3+x^2+x+1 in Z2[x] and F=Z2[x]/I then
    .1)F is an infinite field
    2)F is finite field with 4 elements
    3)F is finite field with 8 elements
    4)F is finite field with 16 elements

    next question
    let G be k group with the generators a and b given by
    G= <a,b:a^4=b^2=1,b2=(a^-1)b>
    if Z(G) denote the center of G then G/Z(G) is isomorphic to
    1)the trivial group
    2)C2,the cyclic group of order 2
    3)C2 x C2
    4)C4

    please please explain the answer these questions were asked by indian institue of technology delhi
    in gate entrance exam
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  2. #2
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    Quote Originally Posted by reflection_009 View Post
    Let I denote the field generated by x^4+x^3+x^2+x+1 in Z2[x] and F=Z2[x]/I then
    .1)F is an infinite field
    2)F is finite field with 4 elements
    3)F is finite field with 8 elements
    4)F is finite field with 16 elements
    Any element in \mathbb{Z}[x]/I has form a + bx + cx^2+dx^3 + (I). Where a,b,c,d \in \mathbb{Z}_2. Furthermore different representations gives different elements. Thus, there are 2\cdot 2 \cdot 2 \cdot 2 = 16 such elements.

    G= <a,b:a^4=b^2=1,b2=(a^-1)b>
    Does you really mean \left< a,b, : a^4 = b^2 = 1, b^2a = a^{-1}b \right> ?
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  3. #3
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    Yes you are right.Yes i mean that. Please thanks alot for your help
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