Gauss Lemma proof：Gauss's Lemma

Gauss's lemma (polynomials)

https://en.wikipedia.org

abstract algebra

https://math.stackexchange.com

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 (number theory)

https://en.wikipedia.org

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

https://crypto.stanford.edu

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' lemma. If R is a UFD

https://www.maths.tcd.ie

We want to prove: Gauss' lemma. If R is a UFD, then R[x] is a UFD. We know that if F is a field, then F[x] is a UFD (by Proposition 47, Theorem 48.

Gauss's Lemma (Polynomial)

https://encyclopedia.pub

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

https://mathweb.ucsd.edu

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)

https://en.wikipedia.org

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 ...