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Math Help - quotient rings 3

  1. #1
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    quotient rings 3

    If R is a finite integral domain, show that R is a field.

    Please show steps. Thanks!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mpryal View Post
    If R is a finite integral domain, show that R is a field.

    Please show steps. Thanks!
    do you know what an integral domain is? do you know what a field is? look at the definitions. you need only show that the set of nonzero elements of a finite integral domain forms a group with respect to multiplication.
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    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by mpryal View Post
    If R is a finite integral domain, show that R is a field.

    Please show steps. Thanks!
    Hi mpryal.

    Pick a nonzero element a in R and consider the set \{a^n:n\in\mathbb Z\}. This must be a finite set of nonzero elements; hence there exist positive integers i,j with i<j such that a^i=a^j. Thus 0_R=a^j-a^i=a^i(a^{j-i}-1_R). Since a^i\ne0_R and R is an integral domain, a^{j-i}-1_R=0_R. \therefore\ a^{j-i}=1_R. \therefore\ a^{j-i-1} is the multiplicative inverse of a in R. (Note that j-i-1\ge0.)
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    Quote Originally Posted by mpryal View Post
    If R is a finite integral domain, show that R is a field.

    Please show steps. Thanks!
    Let R^{\times} = \{r_1,...,r_n\} where r_1 is identity.
    For r\in R^{\times} consider rr_1,rr_2,...,rr_n.
    These must be distinct.
    By pigeonhole it means rr_j = r_1=1 for some j.
    Thus, r has inverse.
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