Re: Irreducible Polynomials that generate primes
> 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 polynomialswith
> 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
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 primesthough 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