• Support PF! Buy your school textbooks, materials and every day products Here!

Irreducible polynomials

  • Thread starter RTH001
  • Start date
  • #1
2
0
If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618


I wouldn't drag imaginary numbers into this. If x^2+1 is reducible over the integers then it would split into two linear factors with integer coefficients. Can you argue why that can't happen?
 
  • #3
2
0


is it because x^2+1 has no real solutions?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
955
Strictly speaking, because the problem said "irreduceable over the integers", it is because there are no integers satisfying [itex]x^2+1= 0[/itex]. Of course, since the integers are a subset of the real numbers yours is a sufficient answer. But your teacher might call your attention to the difference by (on a test, perhaps) asking you to show that [itex]x^2- 2[/itex] is irreduceable over the integers.
 

Related Threads on Irreducible polynomials

  • Last Post
Replies
2
Views
745
  • Last Post
Replies
3
Views
6K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
1
Views
4K
Replies
1
Views
4K
  • Last Post
Replies
12
Views
845
  • Last Post
Replies
2
Views
594
Top