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Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely irreducible if it is irreducible over $\bar{\mathbb F}$, the algebraic closure of $\mathbb F$.

Suppose now that the characteristic of $\mathbb F$ is $2$. Can we characterize the absolutely irreducible polynomials in this case? Another related question of my interest is to know whether there exist absolutely irreducible polynomials of any degree in characteristic $2$. Finally, any reference on absolute irreducibility (in characteristic $2$ or not) and on absolute irreducibility criterion would help.

Some context: I am working on some algorithm on polynomials of characteristic $2$. Until now, I have only been able to work with multilinear polynomials. In order to extend to general polynomials, the idea would be first to factorize my polynomials into (absolutely) irreducible polynomials, and then apply the algorithm to each factor. My problem would be (sort of) solved if the absolutely irreducible polynomials in characteristic $2$ are all multilinear. Even if this is not true (actually, I am not very confident this can be true!), I may be able to use some specific properties of absolutely irreducible polynomials to extend my algorithm to those polynomials.

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Why do you think characteristic 2 is special in this regard? – Mariano Suárez-Alvarez Jul 4 '11 at 23:27
The polynomial $x^n+y\in k[x,y]$ is irreducible over all fields, no? – Mariano Suárez-Alvarez Jul 4 '11 at 23:33
Interesting, it appears that one can test for absolute irreducibility without leaving the initial field, according to [Noether, E., Ein algebraisches Kriterium für absolute Irreduzibilität, Math. Ann. 85, p.26—32 (1922)] – Mariano Suárez-Alvarez Jul 5 '11 at 0:12
Thanks for your comments. Actually, I am working in characteristic 2, so my question was something like "do there exist some specificities of characteristic 2?". Your example rules out the possibility of all absolutely irreducible polynomials to be multilinear. I was convinced there must a simple example, there it is! I'll have a look to Noether's criterion. If somebody has a reference in English, it would be even easier! – Bruno Jul 5 '11 at 0:20

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