Why is x^2 + 1 an Irreducible Polynomial?

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Discussion Overview

The discussion revolves around the irreducibility of the polynomial \(x^2 + 1\) and extends to related polynomials such as \(x^2 + x + 1\). Participants explore the implications of irreducibility in different number systems, particularly the real and complex numbers.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants question why \(x^2 + 1\) is considered irreducible, prompting a discussion on its factors.
  • Others suggest that \(x^2 + 1\) is irreducible over the reals but reducible over the complex numbers, indicating a distinction based on the number system.
  • One participant introduces the polynomial \(x^2 + x + 1\) and asks about the implications of its discriminant on the nature of its roots.
  • A calculation of the discriminant for \(x^2 + x + 1\) is presented, showing that it is negative, which implies no real solutions exist for this polynomial.

Areas of Agreement / Disagreement

Participants appear to agree on the irreducibility of \(x^2 + 1\) over the reals and its reducibility over the complex numbers. However, the discussion regarding \(x^2 + x + 1\) introduces additional complexity, and the implications of its discriminant remain a point of exploration.

Contextual Notes

The discussion does not resolve the broader implications of irreducibility across different fields or the specific conditions under which these polynomials are analyzed.

mathdad
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Why is x^2 + 1 irreducible?
 
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RTCNTC said:
Why is x^2 + 1 irreducible?

What do you think? If you factor this expression, what do you get? Moreover, what could be said about those factors in relation to the real numbers?
 
Complex numbers?
 
Indeed!

In this case we could say that the expression is irreducible over the reals, but reducible over the complex numbers. But I might let someone else chime into give a more formal approach. :)
 
Cool.
 
What about x^2 + x + 1?
 
RTCNTC said:
What about x^2 + x + 1?

What does the discriminant tell you about the roots of this quadratic polynomial?
 
b^2 - 4ac

1^2 - 4(1)(1)

1 - 4

-3

b^2 - 4ac < 0

This means no real solutions.
 

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