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If I have a polynomial with more than one variable, for example $\Bbb C[x,y]$ (with $\Bbb C$ the complex numbers), how can Ii see whether it is irreducible or not? For example, if I have $x^2-y^2-1$, what can I do to know whether it is irreducible or not? In this example I tried to used Eisenstein's theorem in $\Bbb C[x][z]$, but I don't know how to do it. Thanks!

Bernard
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Laura
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    In this case, if you can factor it, you can factor it into first degree polynomials. It shouldn't be too difficult to either find such factors or prove that they cannot exist. But in general there aren't very many simple, powerful tools the way Eisenstein works with a single variable over $\Bbb Z$. – Arthur Oct 29 '20 at 18:48
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    supposes it has a nontrivial factorization -- computing the resultant of your polynomial and the (degree 1) polynomial it purportedly factors with... is a good way to go – user8675309 Oct 29 '20 at 19:56
  • To apply Eisenstein you need to find a prime factor of $y^2+1$ in $\Bbb C[y]$ with multiplicity one. Where did you get stuck on that? – Bill Dubuque Oct 29 '20 at 23:40

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Yes, Eisenstein's criterion is a possibility: rewrite the polynomial as $$X^2-(Y^2+1)\in \mathbf C[Y][X]. $$ $Y+\imath$ is an irreducible factor of all coefficients, but the leading coefficient, and $(Y+ \imath)^2=Y^2-2\imath Y-1$ does no divide the constant term.

Bernard
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  • Sure! Thank you so much – Laura Oct 29 '20 at 19:35
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    Please strive not to add more dupe answers to FAQs. You can quickly locate prior dupes via https://approach0.xyz/search/ besides site search, googling, etc. – Bill Dubuque Oct 29 '20 at 22:52
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    @BillDubuque: I wasn't aware of this link. I'll add it to my bookmarks. Thanks! – Bernard Oct 29 '20 at 23:00
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    @Bernard That's what search is for. Being here almost 6 years and answering over 8000 questions surely you must know that we have many prior instances of such questions. How could you not? – Bill Dubuque Oct 29 '20 at 23:03
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    I know that, but I had no way to check. Also, some questions look like a standard question, but I feel that the O.P. has specific difficulties with the concepts or techniques involved, and I think I can address these specific difficulties. – Bernard Oct 29 '20 at 23:11
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    I just wanted to explain why her intuition about Eisenstein was quite correct. – Bernard Oct 29 '20 at 23:15
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    If need be, you can always add a comment after linking to an (abstract) dupe in order to ensure that the OP knows how to apply the dupe link. By adding yet more dupe answers with no novelty you make it more difficult to locate the" best" answers, and make it more difficult to iteratively refine prior answers (in the hope that they will eventually become "proofs from the book" after enough time being critiqued and improved). – Bill Dubuque Oct 29 '20 at 23:15
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    Yes, but I didn't have myself any dupe to link to, and further, I see links to dupes as useful more for someone who doesn't really know how to start. – Bernard Oct 29 '20 at 23:20
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    DId you even try to search? – Bill Dubuque Oct 29 '20 at 23:21
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    I didn't know your link. – Bernard Oct 29 '20 at 23:22
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    You would if you had searched. Why do you refuse to search? If you did you would better see how the many hundreds of (rushed) dupe answers greatly complicates using the site to locate (the best) answers to questions. – Bill Dubuque Oct 29 '20 at 23:23