Sometimes you want to explain changes you did to a text in (La)TeX, or perhaps to source code, as a coloured diff produced by your favourite VCS, such as git in my case, by attaching it to an email. The net is full of various scripts to do so, with various degree of success; however, all you need is an installation of enscript, something that is standard on most Linux systems, or at least used to be so.
enscript understands a lot of different programming languages, and can highlight them, and not only them, but also various diff files, e.g., something that git normally produces, unified diffs. Now if your diff is saved in a file blah.diff, you can do
enscript -o blah.ps --color -Ediffu blah.diff
to create Postscript file blah.ps with nicely coloured diff. (And you can also produce HTML instead, or convert Postscript to PDF.)
each maximal ideal 
in ![{R=\mathbb{K}[X_1,\dots,X_n]}](https://s0.wp.com/latex.php?latex=%7BR%3D%5Cmathbb%7BK%7D%5BX_1%2C%5Cdots%2CX_n%5D%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
has form 
, for 
, i.e. 
, the ideal of an one-element subset of 
. Its proof in this generality needs quite a bit of commutative algebra. However, if we futher assume that 
) we can give a much quicker proof.![{R=\mathbb{K}[X_1,\dots,X_n]}](https://s0.wp.com/latex.php?latex=%7BR%3D%5Cmathbb%7BK%7D%5BX_1%2C%5Cdots%2CX_n%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. Then there exists 
. To see this, we will show that every 
is algebraic, i.e. a root a nonzero polynomial ![{f\in\mathbb{K}[z]}](https://s0.wp.com/latex.php?latex=%7Bf%5Cin%5Cmathbb%7BK%7D%5Bz%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. Note that the dimension of 
as a vectorspace over 
of the monomials 
under the ring homomorphism 
, and the exponents 
form a countable set. Thus for 
the set
such that 
Thus
![{P\in\mathbb{K}[z]}](https://s0.wp.com/latex.php?latex=%7BP%5Cin%5Cmathbb%7BK%7D%5Bz%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and 
. As 
is linear, i.e. 
maps 
to 
, for 
. As 
, we see that 
, for 
) to this one.
