Register to reply

Irreducibility of Polynomial

by Square1
Tags: irreducibility, polynomial
Share this thread:
Square1
#1
Dec5-12, 05:42 PM
P: 115
Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to
r^3 + 2(r^2)s + (s^3) = 0


I don't understand the next part of the solution. Why introduce this prime number that divides s.
--
Suppose a prime number p divides s.
This implies p divides 2(r^2)s + (s^2)
Or, this implies also the above equals 2(r^2)pa + (p^3)(a^3) = p(2(r^2)a + (p^2)(a^3)) which implies p divides (r^3) which also means p divides r which is contradiction because (r,s) = 1
---
So I follow the above steps, but what does prime number p dividing s or r have anything to do with this?


Next part of solution.
--> If s = 1, (r^3) + 2(r^2) + 1 = 0 means r((r^2) + 2r) = -1 implies r = 1 or -1, but 1 and -1 are not roots, seen by evaluating.
Same argument for s = -1.
This means that r/s is not a root, so not irreducible in Q[x]
---
So I'm lost about this step too. Only thing that I think is that if you let s = 1 or -1, it's as if you are trying to find something about a root in Z...
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Square1
#2
Dec5-12, 06:36 PM
P: 115
sry. i guess this is homework styled question. I don't see a delete button.


Register to reply

Related Discussions
Proving Polynomial Irreducibility over Z Calculus & Beyond Homework 5
Criterion for Irreducibility of a polynomial in several variables? Linear & Abstract Algebra 5
Irreducibility of a polynomial by eisenstein and substitution Calculus & Beyond Homework 2
Irreducibility of a general polynomial in a finite field Calculus & Beyond Homework 3
Algebra question (irreducibility of polynomial) Calculus & Beyond Homework 1