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I solved a question in an exam and I would like to know why do you think I didn't get 100% of this question (they gave me 18/20 points). The question is to state and prove the Eisenstein criterion. I solved in the following manner:

Eisenstein criterion

Let $p(X)=a_nX^n+a_{n-1}X^{n-1}+\ldots+a_0\in \mathbb Z[X]$. If there exist a prime number $p$ such that $p\nmid a_n,p\mid a_i$ for $i=0,\ldots n-1$ and $p^2\nmid a_0$, then $p(X)$ is irreducible in $\mathbb{Z}[X]$.

Proof

Suppose $p(X)$ holds the conditions of the theorem and $p(X)=g(X)h(X)$ with $g(X),h(X)\in \mathbb Z[X], g(X)=b_mX^m+\ldots+b_0$ and $h(X)=c_tX^t+\ldots+c_0$ and $m+t=n$. Since $p\nmid a_n$ and $p|a_i$ for $i=0,1,\ldots,n-1$, there exist $a'\in \mathbb Z_p$ such that $p(X)=a'X^n$ in $\mathbb Z_p[X]$. Since $p(X)=g(X)h(X)$, it follows $g(X)=b'X^m$ and $h(X)=c'X^t$ in $\mathbb Z_p[X]$. Therefore $p^2|b_0c_0$, contradiction because $p^2\nmid a_0$.

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    Points off without an explanation? Lame. – 2017-01-03
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    @KajHansen I'm sorry. I didn't follow you. – 2017-01-03
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    I'm just lamenting the fact that you got points off on a test without an explanation from (presumably) your professor. Seems like common courtesy, lol. Anyways, it might be because you want to also include $m, t > 1$. Otherwise, you can just factor out a constant. – 2017-01-03
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    @KajHansen Yes, so sad. He isn't my professor. Otherwise I could ask him personally why he didn't give me 100% of the question. – 2017-01-03
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    Oops, I meant to write $m, t \geq 1$ above. – 2017-01-03
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    @KajHansen do you think I should have written $m,t>1$ in the beginning of the proof? – 2017-01-03
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    @KajHansen: Why do you want to include $m, t\geq 1$ ? Sometimes a polynomial will be reducible but still not a product of two smaller-degree polynomials. – 2017-01-03
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    Also, to be somewhat pedantic, the criterion needs an $n \geq 1$ requirement. Degree-$0$ polynomials can be reducible. – 2017-01-03

2 Answers 2

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Chances are points were docked for sloppy (or not-perfect) formulation. Here is a bunch of nitpicks.

  • You use the same symbol, $p$, for the polynomial and the prime.
  • The degree of $p$ should be at least $1$, i.e. $n \geq 1$.
  • "Suppose $p(X)$ holds the conditions of the theorem" sounds weird to me.
  • It's not immediately clear what is meant exactly by "Suppose $p(x)$ holds the conditions of the theorem". Being very strict, I would interpret it as "Suppose there exists a prime $p$ satisfying the conditions as in the theorem", but then you haven't formally picked such a $p$.
  • Formally, is your approach to assume that $p(X)$ is reducible and then derive a contradiction, or is your approach to write $p(X)$ as a product $g(X)h(X)$ and deduce that one of $g(X)$, $h(X)$ is a unit? That's not immediately clear at the point that you introduce $g(X)$ and $h(X)$.
  • $a'$ is just (the residue class of) $a_n$; why the "exists $a'$" then?
  • Same for $b'$ and $a'$.
  • The proof is more elegant if you reduce modulo $p$ right at the start without bothering to give names to the coefficients of $g(X)$ and $h(X)$.
  • You could be more explicit about why $p^2 \mid b_0 c_0$ and about the fact that $b_0 c_0 = a_0$.

Now, is this worth deducing $10$% of the points available? I don't know, but there is a case to be made for "the proof is not 100% perfect and therefore it is not worth 100% of points".

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    $m,t>1$ is unnecessary (and cannot be ensured). But $n\geq 0$ is important. – 2017-01-03
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There are two pays to prove Eisenstein's Criterion. One method is to writing out $g(X)$ and $h(X)$ and prove that either $m=n$ or $t=n$. This method is a bit tedious. Second method is to start by taking modulo $p$. This method is about 1 line long. It seems you used a mixture of these two. Your solution is more or less correct but a bit difficult to read. I feel like the grader just skimmed through it, had trouble understanding, and decided to give you 8/10 and not bother fully understanding.

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    Do you know where I can find this proof "1 line long"? do you know exactly where is my mistake? Thank you very much! – 2017-01-03
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    See https://crypto.stanford.edu/pbc/notes/numbertheory/eisenstein.html. It's the first proof on top. You can cut out a lot of the words to make it 1 line. This is essentially what you did half way through. – 2017-01-03