What's the definition of F[x] and F[[x]], where F is a field?

Question

What's the definition of F[x] and F[[x]], where F is a field?

I know this might be a stupid question but I can't seem to find the answer anywhere.

Answer 1

$$ F[x]=\left\{\sum_{i=0}^n a_ix^i\colon a_i\in F,n\in\{0,1,\cdots\}\right\} $$

$$ F[[x]]=\left\{\sum_{i=0}^\infty a_ix^i\colon a_i\in F\right\} $$

Answer 2

$F[x]$ represents the ring of polynomials over the field F. Formally, this ring can be defined as the set of functions with finite support (taking only finitely many nonzero values) from the natural numbers into the field. The operations are defined as follows:

$$ (f+g)(i):= f(i) + g(i) \text{ }\forall f,g \in F[x] \text{ and } i \in \mathbb{N} \\ (fg)(i):= \sum_{j=0}^{i}f(j)g(i-j) \text{ }\forall f,g \in F[x] \text{ and } i \in \mathbb{N} $$

The product defined above is often called Cauchy product.

At this point, you might think that this construction has little to do with the usual polynomials we know from highschool. However, notice first that a function $f$ from $\mathbb{N}$ can be represented as a sequence or infinite tuple as follows:

$$ f=(f(0),f(1),f(2),...) $$

Now, if you define $X^{i}:=(0,0,...,1 (\text{ith-spot}),0,0,0...)$ (i.e. the function sending i to 1 and everything else to zero), you will realize that we have just defined usual polynomials. Checking this might be a good exercise. Prove, for instance, that $XX=X^{2}$. The operations defined above turn quite nicely in just the usual way in which we add an multiply polynomials. For example, the finite support restriction takes care of the fact that polynomials have only finitely many terms.

The ring $F[[X]]$ is called the ring of formal power series over $F$. The definition is identical to the one described above, except for a very important detail: we do not require the functions in $F[[X]]$ to have finite support. Thus, we get a ring whose elements look the way we would expect:

$$ a_{0} + a_{1}X + \dots + a_{n}X^{n} + \dots \text{ with } a_{i} \in F $$

where we have defined $X$ exactly as above.