All Questions
Tagged with polynomials irreducible-polynomials
1,520
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Irreducible polynomial modulo every prime?
There exist irreducible polynomials in $\mathbb{Z}[x]$ (e.g. $x^4-10x^2+1$) which are reducible modulo every prime $p$. (A proof can be found in J.S. Milne's Fields and Galois Theory, page 13.) This ...
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Number of monic irreducible polynomials of prime degree $p$ over finite fields
Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$?
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Reducibility over $\mathbb{Z}_2$?
I have seen that $x^{2}+x+1$ and $x^{4}+x+1$ are irreducible over $\mathbb{Z}_2$ and I thought a polynomial of the form $x^{2^m}+x+1$ for $m\ge3$ would be irreducible too. However using WolframAlpha, ...
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Irreducible homogeneous polynomials of arbitrary degree
Suppose we have an algebraically closed field $F$ and $n+1$ variables $X_0, \dots, X_n$, where $n > 1$. Does there exist an irreducible homogeneous polynomial in these variables of degree $d$ for ...
- 173
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Methods to see if a polynomial is irreducible
Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then ...
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