Irreducibility Multiple Roots

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  Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irreducible over the field F iff it has no roots in F, or Lemma of Theorem: The nonconstant polynomial p(x) an element of F[x] is irreducible over F iff for all f(x), g(

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  This solution shows how to determine irreducibility and use it in a proof of ideal, construct an integral domain and element with given characteristics, complete a proof that a and b are relatively prime, and complete a proof regarding roots of elements

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  According to Eisenstein's Irreducibility Criterion, f(x) is irreducible over the field Q. Irreducible polynomials are investigated. The Eisenstein's irreducibility criterions are examined.

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  Such a polynomial has precisely p^n distinct roots over Z_p, since its derivative is p^n x-1 = -1 (since p^n=0 in Z_p) and is never zero. The set F of roots is a field: 0 and 1 are in F, because 0^{p^n}-0=0, and 1^{p^n}-1 = 0.

• Polynomial Question: Galois group is S7

  So (please see the attached file) is irreducible over (please see the attached file) by the Eisenstein Irreducibility Criterion. We can verify graphically that this polynomial has exactly five real roots.

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  This polynomial is irreducible over Q ( a , b ) , hence ( by Hilbert's irreducibility theorem ) is irreducible over Q for infinitely many specializations of a and b into Q .

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  This solution provides a proof of irreducibility.