Proving Irreducibility of Polynomials over the Integers

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Homework Help Overview

The discussion revolves around proving the irreducibility of the polynomial \(x^2 + 1\) over the integers. Participants are exploring the conditions under which a polynomial can be considered irreducible, particularly focusing on the absence of integer roots.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are examining whether demonstrating the lack of integer roots is sufficient to prove irreducibility. There is a discussion about the implications of factoring polynomials and the relevance of imaginary numbers in this context.

Discussion Status

The conversation is active, with participants questioning the assumptions related to integer solutions and the definitions of irreducibility. Some guidance has been offered regarding the relationship between integer and real solutions, but no consensus has been reached on the best approach to the problem.

Contextual Notes

There is an emphasis on the specific requirement of proving irreducibility over the integers, which may differ from considerations in the real number context. Participants note the potential for differing interpretations based on the wording of the problem.

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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?
 
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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?
 


is it because x^2+1 has no real solutions?
 
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.
 

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