# Help in this proof in Lang's Algebra book (really elementary doubt)

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

• 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 $d$. – Alex Becker Mar 24 '14 at 22:27
• @AlexBecker of course, thanks – user42912 Mar 25 '14 at 2:37