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