Polynomial ring

The polynomials $f(X)=X^{5}+3 X^{4}+X^{3}+4 X^{2}-3 X-1, g(X)=X^{2}+X+1$ can be regarded as belonging to the ring $\mathbb {Z}{}[X]$ or, say, to the ring $\mathbb {Z}_{5}[X]$, depending on how their coefficients are interpreted. Applying the division algorithm with remainder, show that in the first case $f(X)$ is not divisible by $g(X)$, while in the second it is. Is the opposite situation possible?

Prove, using Theorem 3, that if $F$ is a field, then the group of all automorphisms of the ring $F[X]$ that are the identity on $F$ is isomorphic to the group of transformations $X \mapsto a X+b$, where $a, b \in F$ and $a \neq 0$.

Show that a polynomial $f \in F\left[X_{1}, \ldots , X_{n}\right]$ is a form of degree $m$ (see the proof of Theorem 4) if and only if $f\left(t X_{1}, \ldots , t X_{n}\right)=t^{m} f\left(X_{1}, \ldots , X_{n}\right)$, where $t$ is a new variable.

Show that the number of distinct monomials in $n$ independent variables of total degree $m$ equals $\binom {m+n-1}{m}$.

Returning to the definitions of item 1, let us consider the collection $A[[X]]$ of so-called formal power series $f(X)=\sum_{i \geqslant 0} a_{i} X^{i}$ in the variable (indeterminate) $X$, or, if one likes, of sequences $\left(a_{0}, a_{1}, a_{2}, \ldots \right)$ with any, possibly infinite, number of nonzero coefficients $a_{i}$, belonging to a commutative ring $A$. Operations with formal power series from $A[[X]]$ are carried out according to the same rules as operations with polynomials:

Show that the set $A[[X]]$, considered together with these operations, is an associative and commutative ring with identity $1=(1,0$, $0, \ldots )$.

Since arbitrarily high powers $X^{i}$ of the variable $X$ occur in the power series $f=\sum a_{i} X^{i}$, instead of the degree $\operatorname {deg} f$, which no longer makes sense, it is natural to consider the order $\omega (f)$ — an integer equal to the smallest index $n$ for which $a_{n} \neq 0$ (one also sets $\omega (0)=+\infty$).

Show that: i) $\omega (f-g) \geqslant \min \left\{ \omega (f), \omega (g)\right\}$; ii) $\omega (f g) \geqslant \omega (f)+\omega (g)$.

If $A$ is an integral domain, then $\omega (f g)=\omega (f)+\omega (g)$. In particular, together with $A$, the ring $A[[X]]$ is also an integral domain.

Show also that $A[X]$ is a subring of $A[[X]]$.

Polynomials and power series are often used as generating functions of various numerical quantities. We shall explain the meaning of operating with them using two simple examples.