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Background
Notation: ##\Bbb{Q}_{\Bbb{Z}}[x]=\Bbb{Z}+x\Bbb{Q}[x]##
Assumed exercises:
(1)(a) Prove that the only units in ##\Bbb{Q}_{\Bbb{Z}}[x]## are ##1## and ##-1##.
(b) If ##f(x)\in \Bbb{Q}_{\Bbb{Z}}[x]##, show that the only associates are ##f(x)## and ##-f(x)##.
(2)(a) If ##p## is prime in ##\Bbb{Z}##, prove that the constant polynomial ##p## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(b) If ##p## and ##q## are positive primes in ##\Bbb{Z}## with ##p\neq q##, prove that ##p## and ##q## are not associates in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(3)(a) Show that the only divisors of ##x## in ##\Bbb{Q}_{\Bbb{Z}}[x]## are the integers (constant polynomials) and first-degree polynomials
of the form ##\frac{1}{n}x## with ##0\neq n\in \Bbb{Z}##.
(b) For each nonzero ##n\in \Bbb{Z}##, show that the polynomial ##\frac{1}{n}x## is not irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(c) Show that ##x## cannot be written as a finite product of irreducible elements in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
Exercise: Prove that ##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]## if and only if ##p(x)## is either a prime integer or an irreducible polynomial in ##\Bbb{Q}[x]## with constant term ##\pm 1##.
Question:
If I already proved the assumed exercises ##(1)## to ##(3)## above and I want to show one direction for the Exercise above:
##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]## implies ##p(x)## is an irreducible polynomial in ##\Bbb{Q}[x]## with constant term
##\pm 1##
I am not clear on how to show: if an irreducible polynomial
##p(x)\in\Bbb{Q}[x]## is of the form ##p(x)=g(x)x\pm 1,## where ##g(x)\in \Bbb{Q}[x],## then ##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
Thank you in advance
MENTOR note: Replaced all '$' with double '#' for proper mathjax rendering of the page.
Notation: ##\Bbb{Q}_{\Bbb{Z}}[x]=\Bbb{Z}+x\Bbb{Q}[x]##
Assumed exercises:
(1)(a) Prove that the only units in ##\Bbb{Q}_{\Bbb{Z}}[x]## are ##1## and ##-1##.
(b) If ##f(x)\in \Bbb{Q}_{\Bbb{Z}}[x]##, show that the only associates are ##f(x)## and ##-f(x)##.
(2)(a) If ##p## is prime in ##\Bbb{Z}##, prove that the constant polynomial ##p## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(b) If ##p## and ##q## are positive primes in ##\Bbb{Z}## with ##p\neq q##, prove that ##p## and ##q## are not associates in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(3)(a) Show that the only divisors of ##x## in ##\Bbb{Q}_{\Bbb{Z}}[x]## are the integers (constant polynomials) and first-degree polynomials
of the form ##\frac{1}{n}x## with ##0\neq n\in \Bbb{Z}##.
(b) For each nonzero ##n\in \Bbb{Z}##, show that the polynomial ##\frac{1}{n}x## is not irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
(c) Show that ##x## cannot be written as a finite product of irreducible elements in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
Exercise: Prove that ##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]## if and only if ##p(x)## is either a prime integer or an irreducible polynomial in ##\Bbb{Q}[x]## with constant term ##\pm 1##.
Question:
If I already proved the assumed exercises ##(1)## to ##(3)## above and I want to show one direction for the Exercise above:
##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]## implies ##p(x)## is an irreducible polynomial in ##\Bbb{Q}[x]## with constant term
##\pm 1##
I am not clear on how to show: if an irreducible polynomial
##p(x)\in\Bbb{Q}[x]## is of the form ##p(x)=g(x)x\pm 1,## where ##g(x)\in \Bbb{Q}[x],## then ##p(x)## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##.
Thank you in advance
MENTOR note: Replaced all '$' with double '#' for proper mathjax rendering of the page.
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