> 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