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Algebra and analysis questions

  1. Dec 14, 2006 #1
    (1) let f: [a,b] --> R be an integrable function. Consider a sequence (Pn) of tagged partitions with ||Pn|| -> 0. Prove that int over [a,b] of f(x) = lim n-> infinity of S(f, (Pn)).
    - i stated what it means for a function to be integrable, using the episolon-delta definition, but i cant seem to find the step i need.

    (a) Let f: [0,1] -> R be defined by:
    f(x) = 1 if x=1/n for n a natural number
    = 0 otherwise

    is f integrable? if so find int of f(x) over [0,1]
    (b) same question for g: [0,1] -> R
    g(x)=n if x=1/n for some n a natural number
    = 0 otherwise

    (3) prove that a homomorphism from a field to a ring with more than one element must be an isomorphism.
    - i got that kerT, where T is our homomorphism, is trivial. i just dont get the onto proof yet. why is T onto? i tried assuming it's not onto and trying to derive a contradiction that kerT is not trivial.

    check it out:
    suppose T is not onto, then there exists an element r in the ring R such that there does not exist f in the field such that T(f) = r. however, there exists r1 + r2 = r and f1 and f2 such that T(f1)=r1 and T(f2)=r2, then T(f1+f2)=r1+r2=r, which is a contradiction. but this doesnt seem to be right

    (4) prove that Q[x]/<x^2-2> is ring isomorphic to Q[sqrt(2)].
    - i cant find a proper isomorphism. really stuck

    thanks guys!! last homeworks of the semester!
    Last edited: Dec 14, 2006
  2. jcsd
  3. Dec 14, 2006 #2


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    Homework Helper

    1,2. I'm getting sick of saying this, but PLEASE INCLUDE THE DEFINITIONS YOU'RE USING. In this case, how are you defining the integral of a function, and what are the integrable functions?

    3. This isn't true. You always have the trivial homomorphism, sending every element to 0. If you exclude this, you can pretty easily prove the homomorphism is injective, and so is an isomorphism between the field and its image. But there's no way to guarantee it's surjective. For example, say there's a isomorphsim from F->R, and R is a proper subring of S, then there is a non-surjective homomorphsim from F to S. (for a more explicit example, take the inclusion homomorphism from the field Q to the ring R.) Also note your proof doesn't use any properties specific to this case, so essentially you've proven every homomorphism whatsoever is surjective. (by the way, you haven't)

    4. Show more work. Write out definitions.
    Last edited: Dec 14, 2006
  4. Dec 14, 2006 #3
    Question (2) is very simple: do you know the Riemann criterion for integrability? Do you know about Upper and Lower Riemann sums?
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