I normally don't spend my time attempting to disprove basic theorems in elementary linear algebra, but I found myself today not understand why Theorem 10 on Page 137 of Hoffman's (and Kunze... he always gets left in the dust) Linear Algebra is true.
Let $f$ be a non-scalar monic polynomial over the field $F$ and let $f = p_{1}^{n_{1}} \cdot\cdot\cdot p_{k}^{n_{k}}$ be the prime factorization of $f$. For each $j$, $1 \leq j \leq k$, let $f_{j} = \frac{f}{p_{j}^{n_{j}}} = \prod_{i \neq j} p_{i}^{n_{i}}$ Then $f_{1},...,f_{k}$ are relatively prime.
It then proceeds to state, "we leave the easy proof of this to the reader". This result seems blatantly false to me! Since we know that $f = p_{1} p_{2} \cdot \cdot \cdot p_{m}$ if and only if $f$ and f' are relatively prime, where f' denotes the standard derivative of $f$. We then have that, f' = p_{1}' p_{2}p_{3} \cdot\cdot\cdot p_{m} + p_{1}p_{2}'p_{3} \cdot \cdot\cdot p_{m} + \cdot\cdot\cdot + p_{1}p_{2}p_{3} \cdot \cdot \cdot p_{m}' Thus, we assume without loss of generality that $p_{1} \mid f$, then since we know deg$(p_{1}') < $ deg$(p_{1})$, we have that p_{1} \nmid p_{1}'.
Can anyone provide a proof of this result? I'm quite confused as I doubt such an answer would have lasted a good 50 years in reprinting without someone noticing, so I must be wrong.