Let be the set of all polynomials p(x) in the variable x with with integer coefficients. Show that the function defined by is a ring homomorphism
I don't know I am getting confused with the set up on this one. Can some please help me?
you need to show 2 things:
for any two polynomials p(x),q(x) in Z[x]:
g(p(x) + q(x)) = g(p(x)) + g(q(x))
g(p(x)q(x)) = (g(p(x))(g(q(x))
here is something that may help:
if p(x) = a_{0} + a_{1}x + .... + a_{n}x^{n},
then g(p(x)) = p(0) = a_{0} + a_{1}*0 + .... + a_{n}*0 = a_{0} + 0 + .... + 0 = a_{0}.
in other words p(0) returns the constant term of p(x).
a concrete example:
let p(x) = x - 2, and let q(x) = x + 4
then p(x) + q(x) = 2x + 2, and p(x)q(x) = x^{2} + 2x - 8
g(p(x)) = -2
g(q(x)) = 4
g(p(x) + q(x)) = 2
g(p(x)q(x)) = -8.
is it not true that -2 + 4 = 2, and (-2)(4) = -8?
no that is *not* the expression for g(p(x)). your very first equation is wrong. it should be:
the constant term of p(x) (with a coefficients) and q(x) (with b coefficients) is the term not involving any x's.
what you have is g(p(x) + q(x)) = p(x) + q(x), which is not true.
p(0) is always an integer, it's NOT a polynomial.
yes, it's that simple. in other words:
"the constant term of the sum of two polynomials (over Z) is the sum of the constant terms of the two polynomials"
"the constant term of the product of two polynomials (over Z) is the product of the two product terms"
in fact, this statement remains true if we replace the ring Z by any commutative ring R.
even more amazingly, it remains true if we replace p(0) by p(a), for any any element a in our ring R.
the map p(x) → p(a) is called "the evaluation map at a" and is an important ring homomorphism.
people often get confused and think that p(a) is "a polynomial in a". it is not. it is just a ring element obtained by substituting a for x in the expression p(x).
for example, let's pick a = 2 in the examples i gave earlier:
p(x) = x - 2 and q(x) = x + 4.
note that p(a) = 2 - 2 = 0, q(a) = 2 + 4 = 6. these are just numbers, not "polynomials in 2".
now (p+q)(x) = p(x) + q(x) = 2x + 2, so: (p+q)(2) = 2(2) + 2 = 4 + 2 = 6
and (pq)(x) = p(x)q(x) = x^{2} + 2x - 8, so (pq)(2) = 4 + 4 - 8 = 0
note that 0 + 6 = 6, and 0*6 = 0.
we use this homomorphism every time we factor a polynomial:
"high-school style":
solve x^{2} + 2x - 8 = 0 for x.
x^{2} + 2x - 8 = (x - 2)(x + 4) = 0
thus either x - 2 = 0, or x + 4 = 0
x - 2 = 0 means x = 2.
x + 4 = 0 means x = -4. thus the solutions are {2,-4}.
"abstract algebra style"
find the roots of x^{2} + 2x - 8 in Z (a in Z such that a^{2} + 2a - 8 = 0).
since x^{2} + 2x - 8 = (x - 2)(x + 4) applying the homomorphism p(x) → p(a) to both sides, we get:
(a - 2)(a + 4) = a^{2} + 2a - 8 = 0.
since Z is an integral domain, one of a - 2 or a + 4 is 0. thus:
a = a - 2 + 2 = 0 + 2 = 2, or:
a = a + 4 - 4 = 0 - 4 = 4.
do you see the (slight) difference? in ordinary high-school algebra, the shift from a function (a polynomial) to a number (substituting in "something for x") is done without even realizing we do it. in ring theory, this is accomplished explicitly (using a homomorphism), before passing to whatever properties the ring R might possess. we know this is OK, *because* we have a homomorphism.