LEC ALG3 13

Eisenstein’s Criterion

Example 118.1(∂).

Let $f,g$ be polynomials in $R=\mathbb{Q}[x,y,z]$, $f$ and $g$ irreducible polynomials, $\overline{g}$ is irreducible in $R/(f)$. Is $R/(f,g)$ an integral domain?

No, apparently. See Problem 2. If we set $f(x)=x^3-yz$, $g(x)=y^2-xz$, and $h(x)=z^2-x^{2}y$, we have

$$\frac{\mathbb{Q}[x,y,z]}{(f,g,h)}\cong \mathbb{Q}[t^3,t^4,t^5],$$

so $(f,g,h)$ is a prime ideal. Note that $\overline{g(x)}$ is irreducible in $\mathbb{Q}[x,y,z]/f(x)$. We can write $(f,g)=q\cap (f,g,h)$, where $\sqrt{q}$ is prime. This shows that $(f,g)$ cannot be prime.

Theorem 118.2(Eisenstein).

Let $R$ be a UFD, and let $Q$ be its field of fractions. Suppose

$$f(x)=\sum _{i=0}^n{a}_i{x}^i\in R[x]$$

such that $a_n\ne 0$ and $\text{deg}f\ge 1$.

If $p$ is an irreducible element in $R$ such that $p\nmid a_n$, $p|a_i$ for $i=0,\dots ,n-1$, $p^2\nmid a_0$, then $f$ is irreducible in $Q[x]$. If $f$ is primitive, it follows (Theorem 115.10.6) that $f$ is irreducible in $R[x]$.

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[!Proof]-

Let $p$ be as above. Write $f(x)=c(f)f^{\prime }(x)$, where $f^{\prime }(x)$ is primitive. It is sufficient to show that $f^{\prime }(x)$ is irreducible in $Q[x]$. The hypothesis given for $f(x)$ continue to hold for $f^{\prime }(x)$ (easy to show). Thus, we can assume WLOG that $f$ is primitive.

Suppose $f=gh$. Write $g(x)=\sum b_i{x}^i$ and $h(x)=\sum c_i{x}^i$. Suppose $p|a_0$. Then, $p|b_0$ or $p|c_0$. If $p|b_0$, then $p\nmid c_0$, since then $p^2|a_0$. Suppose WLOG $p|b_0$. Then, there exists an integer $k$ such that $p|b_0,\dots ,b_{k-1}$, and $p\nmid b_k$. Note that $k$ cannot be less than $n$, since that would lead to a contradiction (see the expression for $a_n$).

[!Theorem]
Let $R$ be a Noetherian domain. $p$ a prime ideal. If $R$ is a ufd, then every prime ideal has height $1$.