Currently I am learning abstract algebra. The text we are using is Dummit Foote's abstract algebra. It says that the degree formula also holds for multivariable polynomial ring when the coefficient $R$ is an integral domain. Intuitively, this sounds correct, however I am having trouble to formulate a formal proof.
The definition of multivariable polynomial:
The polynomial ring in the varaibles $x_1,x_2,\cdots,x_n$ with coefficients in $R$, denoted $R[x_1,x_2,\cdots,x_n]$ , is defined inductively by $$R[x_1,x_2,\cdots,x_n]=R[x_1,x_2,\cdots,x_{n-1}][x_n]$$
The definition of degree is the following:
A monic term $x^{d_1}x^{d_2}\cdots x^{d_n}$ is called simply a monomial. The exponent $d_i$ is called the degree in $x_i$ of the term and the sum $$d=d_1+d_2+\cdots d_n$$ is called the degree of the term.
The degree formula in the text is
degree $p(x)q(x)$ = degree $p(x)$+degree $q(x)$ if $p(x),q(x)$ are non zero. (Provided $R$ is an integral domain)
It is straight forward to see the multivariable polynomial ring is an integral domain provided $R$ is an integral domain. However, there might be cancellation happens so I don't see how can I prove the degree formula. Can someone give a formal argument on this?
Thanks in advance.