Gauss Lemma proof：Gauss' lemma. If R is a UFD

Gauss's lemma (polynomials)

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abstract algebra

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All versions of Gauss's Lemma lead to the result you are quoting: that primitive polynomials with integer coefficients are irreducible over Z if ...

Gauss's Lemma

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Then f(x) is irreducible in Q[x]. Proof: Suppose not, we will derive a contradiction. Because irreducibility in. Q[x] is unaffected by dividing ...

Gauss's lemma (number theory)

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The proof of the nth-power lemma uses the same ideas that were used in the proof of the quadratic lemma. The existence of the integers π(i) and b(i), and their ...

Number Theory

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Proof: See the last paragraph, and note that ( p 2 − 1 ) / 8 is even when p = ± 1 ( mod 8 ) and odd otherwise. Similarly for ⌊ ( p + 1 ) / 4 ...

Gauss's Lemma (Polynomial)

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Proof: Clearly the product f(x).g(x) of two primitive polynomials has integer coefficients. Therefore, if it is not primitive, there must be a ...

9. Gauss Lemma

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Now we give a way to prove that polynomials with integer coefficients are irreducible. Lemma 9.14. Let φ: R −→ S be a ring homomorphism. 4 ...

Gauss's lemma (Riemannian geometry)

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Gauss' lemma asserts that the image of a sphere of sufficiently small radius in TpM under the exponential map is perpendicular to all geodesics originating at p ...