Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?

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SUMMARY

The polynomial f(x)=x^2-q is irreducible in F_p[x] when p and q are prime numbers, and p is not a quadratic residue mod q. This conclusion holds under the condition that pq=-1 mod 4. The irreducibility can be established through the properties of finite fields and quadratic residues, which are critical in determining the behavior of polynomials over these fields.

PREREQUISITES
  • Understanding of finite fields, specifically F_p
  • Knowledge of quadratic residues and their properties
  • Familiarity with modular arithmetic, particularly mod 4
  • Basic concepts of polynomial irreducibility
NEXT STEPS
  • Study the properties of quadratic residues in finite fields
  • Explore the implications of modular arithmetic on polynomial behavior
  • Learn about irreducibility criteria for polynomials over finite fields
  • Investigate examples of irreducible polynomials in F_p[x]
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Mathematicians, particularly those specializing in number theory and algebra, as well as students studying finite fields and polynomial irreducibility.

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If p and q are prime numbers such that p is not a quadratic residue mod q. Show that if pq=-1 mod 4 then the polynomial f(x)=x^2-q is irreducible in F_p[x].
 
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