If $P(x) \in \mathbb{F}_p[x]$, prime $p$, is monic and irreducible, is $P(x)$ also primitive?
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2In which sense? – Anne Bauval Dec 21 '22 at 15:57
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@DuncanTownsend What is the context? What did you attempted? – Dec 21 '22 at 15:59
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In a word, No!. For example $x^2+1$ is irreducible in $\Bbb{F}_3[x]$, but it is not primitive, because it roots are fourth roots of unity and hence won't generate the multiplicative group of the splitting field. – Jyrki Lahtonen Dec 21 '22 at 16:08
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Some care needs to be used when using the word primitive to describe elemnts/polynomials over finite fields. The other meaning of a primitive polynomial (see @Anne's link) is only interesting when the ring of coefficients has non-trivial divisibility theory, so is uninteresting here when working over a field. – Jyrki Lahtonen Dec 21 '22 at 16:14
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A possibly more common (and more confusing) overloading of the term occurs when people discuss primitive elements carelessly. In the case of finite fields, a primitive element is always a generator of the multiplicative group. Elsewhere in field theory an element is called primitive, if it generates the desired extension field by itself. I elaborated on the difference here. – Jyrki Lahtonen Dec 21 '22 at 16:15
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@AnneBauval thanks for clarifying. I mean it in the "field theory" sense. – Duncan Townsend Dec 21 '22 at 16:32
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Seems that @JyrkiLahtonen has the answer: "no". Thanks for the example! – Duncan Townsend Dec 21 '22 at 16:32