# Problem in the "proof" of Eisenstein's criterion on irreducibility.

I have a problem about a line inside the following proof which is actually from the book Abstract Algebra by Dummit & Foote

Proposition (Eisenstein's Criterion). Let $P$ be a prime ideal in the integral domain $R$ and let $~f(x)=x^n+a_{n-1}x^{n-1}+\dots +a_0$ be a polynomial in $R[x]$ $(n\ge1)$. Suppose $a_{n-1},\dots ,a_0$ are all elements of $P$ and $a_0$ is not an element of $P^2$. Then $f(x)$ is irreducible.

Proof. Suppose $f(x)$ is reducible, say $f(x)=a(x)b(x)$ in $R[x]$, where $a(x),b(x)$ are non-constant polynomials. Reducing this equation modulo $P$ and using the assumptions on the coefficients of $f(x)$ we obtain the equation $x^n=\overline{a(x)b(x)}$ in $(R/P)[x]$, where bar denotes the polynomials with coefficients reduced mod $P$. Since $P$ is prime ideal, $R/P$ is an integral domain, and it follows that both of $\overline{a(x)}$ and $\overline{b(x)}$ have $0$ constant term, i.e, the constant term of both $a(x)$ and $b(x)$ are elements of $P$. But then constant term of $a_0$ of $~f(x)$ as the product of these two would be an element of $P^2$, a contradiction. This completes the proof.

But I cannot understand the line which I have bold in the proof. What I understand is that since $x^n=\overline{a(x)b(x)}$, so comparing the coefficients on both sides we get the product of the constant terms of $~\overline{a(x)}$ and $~\overline{b(x)}$ is zero in $(R/P)[x]$, which is a Integral domain.....then either the constant term of $~\overline{a(x)}=0$ or constant term of $~\overline{b(x)}=0$...But how does both of them is zero?

• Ahh don't you just "love" Dummit and Foote? They just suffocate you with mind-numbing details and write many trivial things, and then they skip steps like this in proofs which give you a headache and make you wonder if you are missing something as obvious as the other details they provide.
– Ovi
Commented Jul 25, 2018 at 22:49

Suppose $b$ mod $P$ has nonzero constant term. Then $a$ mod $P$ must be identically zero, since otherwise its lowest nonzero coefficient would multiply by the constant term in $b$ mod $P$ to produce a nonzero term of degree less than $n$ in the product $ab$ mod $P$, since $P$ is prime. (It wouldn't be cancelled by anything else since all lower degrees of $a$ mod $P$ are $0$.) But that means $a$ is in $P$, and yet the original polynomial is not in $P$, so this can't happen.

• +1 Thanks, was really struggling with this.
– Ovi
Commented Jul 25, 2018 at 23:01

Since $R/P$ is a domain, $R/P[x]$ is a domain. In this ring you have $$x^n = \overline{a(x)} \cdot \overline{b(x)}$$ and $\overline{a(x)},\overline{b(x)}$ have degree $<n$.

Call $$\overline{a(x)} = \sum_{k \le i \le K} \overline{a_i} x^i \\ \overline{b(x)}=\sum_{h \le j \le H} \overline{b_j} x^j$$ Then $$x^n = \left( \sum_{k \le i \le K} \overline{a_i} x^i\right) \cdot \left( \sum_{h \le j \le H} \overline{b_j} x^j\right) = \sum_{m=0}^n \left(\sum_{i+j=m} \overline{a_i b_j} \right) x^m$$

This can be done by recursion: We now have $$a_0b_0\in P$$. By the definition of prime ideal, one of $$a_0$$ and $$b_0$$ must be in $$P$$. Suppose $$a_0\in P$$ but $$b_0\notin P.$$ Then since $$\overline{a(x)}\cdot \overline{b(x)}=\overline{x^n}$$. Then $$a_1b_0+a_0b_1\in P$$. By our assumption, $$a_1b_0\in P$$ since ideals are closed under subtraction. But $$b_0\notin P$$. This forces $$a_1\in P$$. Similarly, $$a_0b_2+a_1b_1+a_2b_0\in P$$ forces $$a_2$$ to be in $$P$$. Proceeding by the same method, then all coefficients of $$a(x)$$ are in $$P$$. Finally, we have $$b_0+b_1a_{m-1}+\cdots +b_ma_0\in P$$ where $$m$$ is the degree of $$a(x)$$.Notice that we let $$b_i=0$$ if $$i>\deg b(x)$$. Anyway, we know that $$a_{m-1},a_{m-2},\cdots,a_0\in P$$. Then this forces $$b_0\in P$$ which contradicts to our assumption.

One idea could be to use the fraction field of $$R/P$$ that we can note $$F_{R/P}$$. We look at $$\overline{x}^n=\overline{f(x)}=\overline{a(x)b(x)}= \overline{a(x)} .\overline{b(x)}$$ in $$F_{R/P}[X]$$ and we directly get that : $$\overline{a(x)} = \alpha.\overline{x}^m$$ and $$\overline{b(x)}=\beta. \overline{x}^l$$ with $$\alpha, \beta \in (F_{R/P})^{\times}$$ and $$m,l\ge 1$$ such that $$n=m+l$$.

Then write for instance $$\alpha = \dfrac{\overline{e}}{\overline{d}}$$ with $$\overline{d}\neq \overline{0}$$. Hence $$\overline{d}.\overline{a(x)}=\overline{e}.\overline{x}^m$$ with $$\overline{d}.\overline{a(x)} \in R/P[X]$$.

Now if $$b_0$$ is the constant term of $$a$$ we then have identifying and using that $$m\ge 1$$ that : $$\overline{d}.\overline{b_0}= \overline{0}$$ and then we deduce that $$\overline{b_0}= \overline{0}$$. Hence $$b_0 \in P$$.

We do the same for $$c_0$$ the constant term of $$b$$.

We then get that $$b_0c_0=a_0 \in P^2$$.