The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial).
When revising some Linear Algebra topics, I got stuck with this exercise. The other part of the exercise was to show that $K[t_1,\dots,t_n]$ is a commutative ring, which was quite easy to see.
For only one variable it was easy to see that the statement is true, as the product of any non-zero polynomials $f$ and $g$ is of degree $\deg(fg)=\deg(f)+\deg(g)$, the maximal coefficient being the product of the maximal coefficients of $f$ and $g$, which must be non-zero since $K$ has no non-trivial zero divisors.
I would like to use a similar argument for the case of $n$ variables, but I do not even know what would be the analogon of the maximal coefficient. I was thinking of the considering those terms with maximal sum of exponents, but of course there might be several such terms, and I do not see why the product (which in this case would also contain several terms with maximal exponent sum) can not be trivial if we multiply all those terms.
 A: Show that if the ring $R$ is an integral domain (has no zero divisors) then $R[x]$ is also an integral domain. This boils down to your observation about the degree of a product of two non-zero polynomials. Then use induction together with the fact (or definition) that $K[t_1, \dots, t_n] \approx K[t_1, \dots, t_{n-1}][t_n]$.
A: You can use any monomial ordering, i.e. any total ordering $<$ on monomials such that (boldletters denote multi-indeterminates and multi-exponents):
(i) $\;1<\boldsymbol{x^m}$ for any monomial $\boldsymbol{x^m}\neq 1$
(ii) If $\;\boldsymbol{x^m}<\boldsymbol{x^n}$, then for any monomial $\boldsymbol{x^p}$, we have  $\;\boldsymbol{x^m}\,\boldsymbol{x^p}<\boldsymbol{x^n}\,\boldsymbol{x^p}$.
Some examples of such monomial orderings are obtained choosing an order on the indeterminates first, say $x_1>\dots>x_n$, then we can use:


*

*the lexicographical order,

*the graded lexicographical order: order monomials first by total degree, then use lexicographical order for monomials of the same degree,

*the reverse graded lexicographical order: order monomials first by total degree, then use reverse lexicographical order for monomials of the same degree.

