Polynomials f(x) & g(x) in Z[x] Relatively Prime in Q[x]

  • Context: Graduate 
  • Thread starter Thread starter esisk
  • Start date Start date
  • Tags Tags
    Polynomials
Click For Summary
SUMMARY

Polynomials f(x) and g(x) in Z[x] are relatively prime in Q[x] if and only if the ideal they generate in Z[x] contains an integer. The proof establishes that if f(x) and g(x) are coprime in Q[x], there exist polynomials a(x) and b(x) in Q[x] such that a(x)f(x) + b(x)g(x) = 1. Consequently, an integer n exists such that n a(x) and n b(x) have coefficients in Z, leading to the conclusion that the ideal generated by f and g in Z[x] contains an integer.

PREREQUISITES
  • Understanding of polynomial rings, specifically Z[x] and Q[x]
  • Knowledge of ideals in ring theory
  • Familiarity with the concept of coprimality in polynomials
  • Basic grasp of coefficients in polynomial expressions
NEXT STEPS
  • Study the properties of polynomial rings Z[x] and Q[x]
  • Learn about ideals and their generation in ring theory
  • Explore the implications of coprimality in polynomial equations
  • Investigate the relationship between polynomial coefficients and integer solutions
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying polynomial theory and its applications in number theory.

esisk
Messages
43
Reaction score
0
trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework
 
Physics news on Phys.org
Z[x] contains an integer => f(x), g(x) coprime in Q[z] is easy. If f(x) and g(x) are coprime in Q[x], then there exist a(x) and b(x) in Q[x]. such that a(x) f(x) + b(x) g(x) = 1. There is an integer n such that n a(x) and n b(x) have coefficients in Z (think about why this is). Then n a(x) f(x) + n b(x) g(x) = n, so the ideal generated by f and g in Z[x] contains an integer.
 
Last edited:
This is quite Rochfor1, thank you. And, yes, I will be able to do the other implication.
 
Sorry, I meant "quite clear". Thanks
 

Similar threads

Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
5K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Irreducible polynomials and prime elements
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Proving properties of polynomial in K[x]
  • · Replies 8 ·
Replies
8
Views
2K
MHB Proving Z[x] and Q[x] is not isomorphic
  • · Replies 5 ·
Replies
5
Views
3K
MHB Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Confused about Gauss's Lemma of Irreducibility
  • · Replies 4 ·
Replies
4
Views
3K
MHB The irreducible polynomial is not separable
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Generators of Galois group of ## X^n - \theta ##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K