Is Eisenstein's Criterion Applicable to Polynomials in the Gaussian Integers?

[Ted123]

My statement of Eisenstein's criterion is the following:

Let $R$ be an integral domain, $P$ be prime ideal of $R$ and $f(x) = a_0 + a_1x + ... + a_n x^n \in R[x]$.

Suppose
(1) $a_0 , a_1 , ... , a_{n-1} \in P$
(2) $a_0 \in P$ but $a_0 \not\in P^2$
(3) $a_n \not\in P$

Then $f$ has no divisors of degree $d$ such that $1\leqslant d \leqslant n-1$. In particular if $f$ is primitive and (1)-(3) hold then $f$ is irreducible.

I would like to see an example of how we can use this criterion in the Gaussian integers $R= \mathbb{Z}$.

I know $1+i$ is a Gaussian prime so can I anyone give me a quick example of a polynomial with coefficients in $\mathbb{Z}$ and how to use this criterion?

I know how to use it when $R=\mathbb{Z}$, for example to show $f(x)= x^2 -1 \in \mathbb{Z}[x]$ is irreducible we just check that for a prime $p$: $p | a_n$, $p | a_i$ for all $i<n$ and $p^2 \not | a_0$ . I'm confused as to how to use the version of it above for a polynomial in $\mathbb{Z}<i>[x]</i>$.

 

[DonAntonio]

My statement of Eisenstein's criterion is the following:

Let $R$ be an integral domain, $P$ be prime ideal of $R$ and $f(x) = a_0 + a_1x + ... + a_n x^n \in R[x]$.

Suppose
(1) $a_0 , a_1 , ... , a_{n-1} \in P$
(2) $a_0 \in P$ but $a_0 \not\in P^2$
(3) $a_n \not\in P$

Then $f$ has no divisors of degree $d$ such that $1\leqslant d \leqslant n-1$. In particular if $f$ is primitive and (1)-(3) hold then $f$ is irreducible.

I would like to see an example of how we can use this criterion in the Gaussian integers $R= \mathbb{Z}$.

I know $1+i$ is a Gaussian prime so can I anyone give me a quick example of a polynomial with coefficients in $\mathbb{Z}$ and how to use this criterion?

I know how to use it when $R=\mathbb{Z}$, for example to show $f(x)= x^2 -1 \in \mathbb{Z}[x]$ is irreducible we just check that for a prime $p$: $p | a_n$, $p | a_i$ for all $i<n$ and $p^2 \not | a_0$ . I'm confused as to how to use the version of it above for a polynomial in $\mathbb{Z}<i>[x]</i>$.

What about the polynomial $\,x^2+(1+i)\in\left(\mathbb{Z}<i>\right)[x]\,</i>$ ?

BTW, in your example, $\,x^2-1\in\mathbb{Z}[x]\,$ is reducible...;>)

DonAntonio

 

[Ted123]

What about the polynomial $\,x^2+(1+i)\in\left(\mathbb{Z}<i>\right)[x]\,</i>$ ?

BTW, in your example, $\,x^2-1\in\mathbb{Z}[x]\,$ is reducible...;>)

DonAntonio

Is the polynomial $$f(x) = x^7 + (3-i)x^2 + (3+4i)x + (4+2i) \in \mathbb{Z}<i>[x]</i>$$ irreducible?

$2+i$ is a Gauassian prime isn't it? And 2+i does not divide 1, 2+i | 3-i , 2+i | 3+4i , 2+i | 4+2i and (2+i)^2 = 3+4i which does not divide 4+2i.