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My doubt is really elementary:

Let $(f_1,\ldots,f_n)$ be a row matrix with some component with leading coefficient $1$ and $d$ the smallest degree of a component of $f$ with leading coefficient $1$.

Why can we assume up to elementary row operations that $f_1$ has leading coefficient $1$, degree $d$, and that $\deg f_i\lt d$ for $j\neq 1$?

I know that we can assume $f_1$ has coefficient $1$, degree $d$, but why does $\deg f_i\lt d$ for $j\neq 1$? I'm trying to use euclidean algorithm, dividing the $f_i, i\neq 1$ by $f_1$, the problem is the remainder of the division, I need help.

Source of the doubt (Lang's Algebra page 847):

Thanks in advance

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    $\begingroup$ Using elementary row operations, we can first pivot the smallest monic polynomial so that it is $f_1$. Elementary row operations are sufficient to perform the Euclidean algorithm, so we can replace each $f_j$ for $j\ne 1$ with its remainder mod $f_1$, which has degree less than $d$. $\endgroup$ – Alex Becker Mar 24 '14 at 22:27
  • $\begingroup$ @AlexBecker of course, thanks $\endgroup$ – user42912 Mar 25 '14 at 2:37

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