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Thread: Field

  1. #1
    Jan 2011


    if R is an integral domain and K is sub-ring of R. if R is considered as a finite dimensional vector space over the field K then R is a field.
    is it possible to prove this statement.
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  2. #2
    MHF Contributor

    Mar 2011

    Re: Field

    yes it is. it suffices to prove that any non-zero element in R is a unit, since R is already known to be a commutative ring with identity (unity). moreover, we only have to prove that non-zero elements of R-K are units, since K is known to be a field (and hence any non-zero element of K has a multiplicative inverse in K).

    so pick a in R - K. and suppose that dimK(R) = n.

    consider the set {1,a,a2,...,an}. this has n+1 elements, so must be a linearly dependent set.

    therefore there are c0,c1,...,cn in K, not all 0, such that:

    c0+c1a+...+cnan = 0.

    hence the ideal in K[x], J = {f in K[x]: f(a) = 0} is non-zero. since K is a field, K[x] is a principal ideal domain, thus:

    J = (m(x)), for some polynomial m(x) in K[x].

    i claim m(x) is irreducible over K. for suppose that m(x) = h(x)k(x) where deg(h),deg(k) < deg(m) (so that neither h(x), nor k(x) is a unit).

    then we have 0 = m(a) = h(a)k(a), since R is an integral domain. suppose h(a) = 0 (the proof if k(a) = 0 is similar).

    then (m(x)) is contained in (h(x)), and this containment is strict, because h(x) is not in (m(x)):

    if h(x) = r(x)m(x) = r(x)h(x)k(x), then (since K[x] is also an integral domain, since K is a field, and thus an integral domain)

    1 = r(x)k(x), contradicting the fact that k(x) is not a unit. but, by definition, h(x) is in J, since h(a) = 0.

    since h(x) cannot be both in J and not in J, this contradiction shows m(x) must be irreducible.

    write m(x) = d0 + d1x +...+ drxr.

    i claim d0 ≠ 0, since m(x) is irreducible (otherwise x divides m(x)).

    since m(a) = 0, we have d0 + d1a +...+ drar = 0.

    thus 1 = -(d1/d0 + (d2/d0)a +...+ (dr/d0)ar-1)a,

    which shows that a is a unit, hence R is a field.
    Thanks from saravananbs
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