Irreducible polynomial of degree 3 in Z5

Asked Jun 16 '21 at 23:01

Active Jun 17 '21 at 01:44

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I am trying to find an irreducible polynomial of degree 3 in Z5 and then use that polynomial to construct a finite field.

I found that the polynomial $x^3 + x^2 + 1$ is irreducible in Z5 (since none of 0, 1, 2, 3,4 are roots). But I'm stuck on how to use that polynomial to construct a finite field.

edited Jun 17 '21 at 01:44

RobPratt

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asked Jun 16 '21 at 23:01

acme_2020

1

wouldn't $-1 \equiv 4 \pmod{5}$ be a root? – gt6989b Jun 16 '21 at 23:09
Notice that $4^3 + 1 = 0$ in $\mathbb{Z}_5$. $x^3+1$ is not irreducible over $\mathbb{Z}_5$ – Adler Marques Jun 16 '21 at 23:09
@AdlerMarques thanks, I have edited my question – acme_2020 Jun 16 '21 at 23:20
It looks like $1^3+1^2+1+2\equiv 0\mod 5$... – abiessu Jun 16 '21 at 23:21
@abiessu I think x^3 + x^2 + 1 works – acme_2020 Jun 16 '21 at 23:33
Um, but now you need to check $3$ and $4$, too... – paul garrett Jun 16 '21 at 23:43
@paulgarrett I have checked 3 and 4 and they aren't roots either, I forgot to type them above in the question – acme_2020 Jun 16 '21 at 23:44
I am still stuck on constructing the finite field – acme_2020 Jun 16 '21 at 23:45
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Here's an article I wrote about it in detail. – MJD Jun 17 '21 at 00:50
@MJD thanks, would the field be F5? – acme_2020 Jun 17 '21 at 03:01
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The field you'll get from a degree-3 irreducible polynomial will be $F_{5^3}$. – MJD Jun 17 '21 at 05:10

Irreducible polynomial of degree 3 in Z5

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