a question for the irreducibility test for degree 2 or 3 polynomial

Asked Mar 03 '19 at 16:33

Active Mar 10 '19 at 10:35

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I have a question for the irreducibility test for degree 2 or 3 polynomial. The test states: "Let F be a field, f(x) ∈ F[x], and degf(x) = 2 or 3. Then f(x) is reducible over F if and only if f(x) has a zero in F."

In the textbook, it states the test can be applied to polynomials in Z2[x] or Z3[x]. However, I don't think Z2 and Z3 are fields, just like Z is not field. Can anyone help? Thanks!

edited Mar 10 '19 at 10:35

Sil

16,612

asked Mar 03 '19 at 16:33

RandomThinker

3

Wrong! Z2 and Z3 are fields, if this means $\mathbb Z/2\mathbb Z$, respectively $\mathbb Z/3\mathbb Z$. – user26857 Mar 03 '19 at 16:34
2

For a prime $p,$ the ring $\mathbb Z / p \mathbb Z$ is a field. The reason there are multiplicative inverses: if $n$ is an integer not divisible by $p,$ there are integers $x,y$ such that $nx-py = 1,$ or $nx= 1+py,$ finally $nx \equiv 1 \pmod p$ – Will Jagy Mar 03 '19 at 16:37
I see! Thanks guys! – RandomThinker Mar 03 '19 at 16:49
See this duplicate about $\Bbb Z/n$. – Dietrich Burde Mar 03 '19 at 17:49

a question for the irreducibility test for degree 2 or 3 polynomial

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