Abstract Algebra | Eisenstein's criterion

We present a proof of Eisenstein's criterion along with some examples.

[felipelopes]

Damn, these proofs were amazing. Keep these videos coming!

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I believe that treating Gauss's Lemma as a lemma is really insulting. This fact after 30 seconds of meditation tells us that R[x] is an UFD for any UFD R. This is a very good result

[digxx]

Can anyone help me out here at 15:10. p(x) is a rational polynomial. Let's name the coefficient of x^n as c_n/d_n. Who says that d_n does not contain the prime p in which case p(x) would not be defined over Z_p[x] or put differently d/d_n in d/c*p(x) does not allow to factor out the prime p.

[otakurocklee]

But Gauss' lemma (as you've proved it here) only applies to monic polynomials. How can you deduce that a non-monic polynomial irreducible over integers is also irreducible over the rationals?

[cowworker]

i found many proofs for eisenstein criterium, and this one is simple and convincing. But i still have a question, if you could help me.
At 21:19, we see that degree of C(X) equals n. Thus degree of B(X) is 0. But if b_0 is different from 1 and -1, B(X)=b_0 is not inversible in Z[X], so b_0 * C(X) is a reduction i think (like, for example, f(X)=2X is factorised in 2 times X in Z[X]). Isn't it right? thanks

[brandonk]

I'm struggling with a counter example to both approaches: x^2+4x+16. By Eisenstein, p=2 (or using 2^2), it should be reducible. If we use Z_3 where p=3 x^2+x+1 shows it should be reducible when x=1 a factor. The original roots are complex to boot. If I introduce x=x+1, then it changes to x^2+6x+21. Then using p=3 satisfies Eisenstein. Is there a way to know when to not trust the false positive given by Eisenstein and the Zp test and/or which element to use to test?