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Lemma Let $a(x) = b(x)c(x)$, where $a(x), b(x)$, and $c(x)$ have integer coefficients. If a prime number p divides every coefficient of $a(x)$, it either divides every coefficient of $b(x)$ or every coefficient of $c(x)$.

PROOF: If this is not the case, let $b_r$ be the first coefficient of $b(x)$ not divisible by $p$, and let $c_t$ be the first coefficient of $c(x)$ not divisible by $p$. Now, $a(x) = b(x) c(x)$, so

$\qquad a_{r + t} = b_0c_{r + t} +\cdots+ b_rc_t + \cdots + b_{r + t}c_0$

Each term on the right, except $b_rc_t$ is a product $b_ic_j$ where either $i > r$ or $j > t$. By our choice of $b_r$ and $c_t$, $\color{red}{\text{if }i > r\text{ then }p\,|\,b_i,\text{ and }j > t\text{ then }p\,|\,c_j}$. Thus, $p$ is a factor of every term on the right with the possible exception of $b_rc_n$ but $p$ is also a factor of $a_{r + t}$. Thus, $p$ must be a factor of $b_rc_n$ hence of either $b_r$ or $c_n$ and this is impossible.

Is there a typo at where it is highlighted with red in the proof?

The greater than sign ">" should be replaced with less than sign "<" to become:

$\qquad$ "if $i < r$ then $p\,|\,b_i$, and $j < t$ then $p\,|\,c_j$"

Correct?

Community
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hchar
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    Yes, this is indeed a typo. – Matemáticos Chibchas Feb 17 '20 at 02:55
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    The "either $i>r$ or $j>t$" of the previous line should also be changed to "either $i \leq r$ or $j \leq t$". – Eric Towers Feb 17 '20 at 02:57
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    See this answer for this proof presented more visually, and also more clearly. The highlighted text is correct since the indices are as normal $, b(x) = \sum b_k x^k ,$ and "first" means "leading", i.e. $,b_r x^r,$ is the leading term when reduced $!\bmod p,,$ which is a much clearer way to view it - as I explain in the link. – Bill Dubuque Feb 17 '20 at 03:01

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Part of the issue is that "first coefficient" can be interpreted in more than one way. It seems the author interprets it as the first one found when you write out the polynomial in order from highest to lowest powers and then check each term, so it'll give the coefficient of the highest power found. However, it can also mean the coefficient of the lowest power, which is your interpretation.

With the author's apparent interpretation, the statements are correct. However, if you wish to use your interpretation, then you are correct about needing to change that line. Also, similar to what Eric Towers's comment states, you should then also change the previous line to "either $i \lt r$ or $j \lt t$".

John Omielan
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