Question

Let R be a UFD. For any irreducible p in R, consider the quotient map \(f_p:R\rightarrow R/pR\) and extend this to a homomorphism \(f_p: R[x]\rightarrow (R/pR)[x]\) defined by \(f_p(\sum_i (a_i)x^i =\sum_i f_p(a_i)x^i\)
a) Show that a polynomial h(x) is in the kernel of \(f_p\) if p is a common divisor of the coefficients of h(x)
b) Show that g(x) in R[x] is primitive iff for all irreducible p, \(f_p(g(x))\neq 0\)
still have c and d, but later on, after solving a) and b) I will post
Please, help

Answer 1

@jtvatsim

Answer 2

Well I can say right off the bat that I have not gotten to this part of abstract yet, but I'm looking into my textbook now. :)

Answer 3

@Kainui

Answer 4

I got a

Answer 5

ok, I post c, d if someone can help, please
c) Show that (R/pR)[x] is integral domain for all irreducible p.
d) Conclude that if h(x) and g(x) are primitive in R[x], then h(x)g(x) is primitive as well.
I got d also

Answer 6

So, remain b, c

Answer 7

Yup, I am stumped. The vocabulary and notation is throwing me off, but it gives me something to "look forward to/dread" once I get to this part of abstract. :)

Answer 8

Thanks anyway!! :)

Answer 9

eating!! brb

Answer 10

I got b