I have been working on the following problem:
Let $A$ be a ring containing a field $F$. If $a_1 ,\dots, a_n \in A$, show that there is a unique ring homomorphism $\varphi : F[x_1,\dots,x_n] \to A$ with $\varphi(x_i)=a_i$ for each $i$.
I have already thought enough about this question and I do not know how to express a polynomial which has $n$ variables in a summation. One knows that if it is a polynomial in $x$, then $f(x)=\sum_{i=0}^{n} a_i x^i$, but if $f(x_1,\dots,x_n)$ is a polynomial in $n$ variables then how can I express it in the form of a summation?
You can write it in a few ways.
You can do it iteratively: $$\sum_{k_1=0}^{m_1}\cdots\sum_{k_n=0}^{m_n}b_{k_1}^1\cdots b_{k_n}^{m}x_1^{k_1}\cdots x_n^{k_n}$$ Or you can list the terms by degree (which is useful in some scenarios): $$\sum_{k=0}^{m}\left(\sum_{k_1 + \cdots + k_n = k}b_{k_1}^1\cdots b_{k_n}^{m}x_1^{k_1}\cdots x_n^{k_n}\right)$$