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Suppose $F_{p^k}$ is a finite field. If $F_{p^{nk}}$ is some extension field, then the primitive element theorem tells us that $F_{p^{nk}}=F_{p^k}(\alpha)$ for some $\alpha$, whose minimal polynomial is thus an irreducible polynomial of degree $n$ over $F_{p^k}$.

Is there an alternative to showing that irreducible polynomials of arbitrary degree $n$ exist over $F_{p^k}$, without resorting to the primitive element theorem?

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Yes. See this for the case $k=1$. The same argument works when $k>1$. I like to think that the more general version has been covered here. The Möbius inversion formula for the number of irreducible polynomials works out irrespective of whether $k=1$ or not, and the linked question explains, how to deduce that the number is always positive. –  Jyrki Lahtonen Jan 27 '13 at 15:35
    
Alternatively you can use the argument of the linked question to prove the existence of an irreducible polynomial of degree $nk$ over the prime field. The factors of that polynomial (over the extension field) are then necessarily of degree $n$. –  Jyrki Lahtonen Jan 27 '13 at 15:39
    
Does the fact that the multiplicative group is cyclic count as a use of the primitive element theorem? –  Lior B-S Jan 27 '13 at 18:00
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For $n=2$ and $n=3$ there exists a truly elementary proof, which is unfortunately hard to generalize... –  N. S. Jan 27 '13 at 18:34

1 Answer 1

up vote 4 down vote accepted

A very simple counting estimation will show that such polynomials have to exist. Let $q=p^k$ and $F=\Bbb F_q$, then $X^{q^n}-X$ is the product of all irreducible polynomials over $F$ of degree dividing $n$. The degree of the product of all irreducible polynomials over $F$ of degree strictly dividing $n$ can then be estimated as at most $$ \sum_{d\mid n, d\neq n}q^d\leq\sum_{i<n}q^i=\frac{q^n-1}{q-1}<q^n=\deg(X^{q^n}-X), $$ so they cannot account for all the irreducible factors of $X^{q^n}-X$.

I should add that by starting with all $q^n$ monic polynomials of degree $n$ and using the inclusion-exclusion principle to account recursively for the reducible ones among them, one can find the exact number of irreducible poltnomials over $F$ of degree $n$ to be $$ \frac1n\sum_{d\mid n}\mu(n/d)q^d, $$ which is a positive number by essentially the above argument (since all values of the Möbius function $\mu$ lie in $\{-1,0,1\}$ and $\mu(1)=1$). A quick search on this site did turn up this formula here and here, but I did not stumble upon an elementary and general proof not using anything about finite fields, although I gave one here for the particular case $n=2$. I might well have overlooked such a poof though.

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