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Re: Irreducible Polynomials that generate primes

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  • Dick
    Hello, ... The ... i.e. ... with ... Yes, thank you, I retract my previous assertion that the polynomial in question is reducible. Of course, altering the
    Message 1 of 5 , Jun 10, 2005
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      Hello,

      > Dick, I think you just do not know the definition of irreducible.
      The
      > polynomial example x^2+x+2 is indeed irreducible over the integers;
      i.e.
      > it can not be written as the product of two non-trivial polynomials
      with
      > integer coefficients.

      Yes, thank you, I retract my previous assertion that the polynomial
      in question is reducible. Of course, altering the definition to
      allow consideration of zero degree polynomials would legitimize the
      operation of dividing through by a common factor as a "reduction".

      To explicitly say this operation is not a "reduction" kind of
      squeezes the life out of the word don't you think?

      My careless and liberal abuse of the definition is certainly more
      natural and intuitive, but of course we must keep established
      definitions as they are, otherwise all hell would break loose.

      Even the definition Chris has given here could still apply depending
      on the definition of "non-trivial polynomial", for I could justify
      the notion that a polynomial of 0 degree (i.e. - a constant) is
      trivial only if f=0 or f=1. Then a polynomial whose solution set
      shares a common factor would be reducible by the standard
      definition. Which to me is a clearer, more comprehensive way of
      looking at it, at least as it applies to this problem.

      Obviously, math is full of definitions where a choice had to be made
      one way or another and usually will go the way that best supports the
      problem under study at the time. Anyone know the historical
      underpinnings of the definition of irreducibility and whether this
      issue of zero degree polynomials has come up before. i.e. - why are
      all polynomials of zero degree necessarily trivial? Obviously by
      definition, but the definition could have gone another way, a way
      that in some applications makes a problem simpler and easier to state
      and/or comprehend.

      I'd be interested in knowing the how, when, who and why surrounding
      the definition in it's current, accepted form. Perhaps it stems from
      work on a math problem where this "alternate" definition wreaks
      havoc, or perhaps the alternate actually streamlines the original
      problem as well, as it seems to do here.

      Note that I do not promote or suggest "fixing" anything, I just find
      it interesting how these things go down. The definition of the gamma
      constant is another good example of one where intimacy with the
      problem suggests a more "natural" and logical definition that is
      slightly different from the one which became accepted - Anyone care
      to guess what I'm talking about on the gamma constant?

      > Your analysis of how to guess a poly has infinitely many primes
      though was
      > correct--you look at it modulo each of the primes.

      It's really interesting how simple and clear this is in my mind.
      Obviously, what I'm doing is equivalent to looking at a poly modulo
      each of the primes, but my method and reasoning is not *directly*
      looking at it modulo each of the primes, it is only equivalent to
      that. What I'm actually doing is a different way of looking at
      things.

      Regards

      -Dick Boland
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