Proving Irreducibility of x^4 −7 Using Polynomial Theorems

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SUMMARY

The polynomial x^4 − 7 is proven to be irreducible over the field of rational numbers Q. This conclusion is established using Eisenstein's Criterion, which confirms that the polynomial does not factor into lower-degree polynomials with rational coefficients. The discussion highlights the application of this theorem specifically to polynomials of the form x^n − p, where p is a prime number, reinforcing the irreducibility of x^4 − 7.

PREREQUISITES
  • Understanding of polynomial functions and their properties
  • Familiarity with Eisenstein's Criterion for irreducibility
  • Knowledge of rational numbers and fields
  • Basic concepts of algebraic number theory
NEXT STEPS
  • Study Eisenstein's Criterion in detail to understand its applications
  • Explore other irreducibility tests such as the Rational Root Theorem
  • Investigate the implications of irreducibility in algebraic number theory
  • Learn about polynomial factorization techniques over different fields
USEFUL FOR

Mathematicians, algebra students, and educators interested in polynomial theory and irreducibility proofs will benefit from this discussion.

Poirot1
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Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.
 
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Which theorems related to irreducibility do you know?
 

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