Cannot understand a Finite field textbook example

Question

I am reading a book that has the following text which I don't understand.

Let $F$ be a finite field and $\alpha \in K$ where $K$ is an extension of $F$. Then we write $F[\alpha]$ to indicate all sums of the form $\sum x_i \alpha^i$ where $x_i \in F$ and where all but a finite number of the coefficients $x_i$ are zero.

Example 2.31 (i) For the real numbers $\mathcal{R}$ and $i^ 2 = −1$, we have that $\mathcal{R}[i]$ is all finite sums of the form $\sum x_k i^ k$ . We can use the properties that $i ^3 = −i$, $i^ 4 = 1$, and so forth, to reduce any sum of this form to a single complex number $a + bi$, so elements of $\mathcal{R}[i]$ are just the complex numbers.

What are the limits of the sum? Does that mean $F[\alpha] =\sum_{\forall x_i \in F} x_i \alpha^k$ ?

It says $x_i \in F$ but $\mathcal{R}$ is infinite. Is it an infine sum? Then how can one reduce that into a single complex number?

What is the name of such notation? If I try to search for it what do I call that?

Answer 1

You are right that the example does not match what is introduced before, because the real numbers are not a field extension of any finite field.

That being said, a standard name for that object $F[\alpha]$ would be the ring extension of $F$ by the element $\alpha$ (inside a given extension $K$ which contains $\alpha$). Or equivalently, this is the smallest subring of $K$ which contains $F$ and $\alpha$. In case $F$ is a field and $\alpha$ is algebraic over $F$ (like in the example in your book with $F = \mathbb R$ and $\alpha=i$), this is also the same as the field extension of $F$ by the element $\alpha$, or equivalently, the smallest subfield of $K$ which contains $F$ and $\alpha$. The element $\alpha$ would then be called a primitive element of that extension.

Another way to think of it is that $F[\alpha]$ is the image, inside $K$, of the abstract ring of polynomials $F[X]$, under the homomorphism which sends $X$ to $\alpha$ ("evaluate the polynomial at $\alpha$").

Nowhere in any of these definitions does one need that $F$ is finite.

What is finite is the number of non-zero coefficients in those sums (as in polynomials).

Answer 2

The index $i$ in the sum goes from $0$ to $\infty$ in the natural numbers but there is a restriction written right after, that $x_i=0$ for all but finitely many indices. So we only deal with finite sums.
These finite sums are the polynomials 'in $\alpha$', these are the elements of $F[\alpha]$, and it is a subring of $K$.

If $F[X]$ denotes the ring of polynomials over $F$, then $F[\alpha]$ is the set of substitutional values writing $\alpha$ in place of $X$, i.e. $$F[\alpha]\ =\ \{p(\alpha)\,:\,p\in F[X]\}\,.$$