Cannot understand a Finite field textbook example

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?

• If the field is finite and the topology is Hausdorff, the topology is discrete. Sequences converge if and only if they are eventually constant. May 24 at 22:37
• @RodrigodeAzevedo I didn't know what this thing is called. In the question I ask for a name of the notation. Please suggest an informative title. May 25 at 12:46
• @NeelBasu I don't know anything about finite fields. What I do know is that "I don't understand this" says very little about the content of the question, which is unfortunate. The title should tell enough for people to decide whether to click on this question or not. May 25 at 12:58

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).

• It says "elements of $\mathcal{R}[i]$ are just the complex numbers." that means it is not a single value but a set. But a summation (even if it is infinite) yields a single value. not a set. What does this "all sums" mean ? Does it mean All $\alpha^{i}$ for each $x_{i}$ or all $x_{i}$ for each $\alpha^{i}$ ? May 25 at 12:44
• Sure it's a set. It's the image of all polynomials, a.k.a. all (finite) sums of powers of $\alpha$ with coefficients from $F$. In the case of real numbers and $i$, this contains e.g. $3+12.4i$, and $-\sqrt[17]{82.3}+999i+0.6i^2+3i^3+5i^4+2.331i^5+6\pi i^6$, and $52$, and $-8i^3+500 i^{829}-\sqrt{\pi} i^{100001}$, and ... many more. You will see these are all complex numbers, and every complex number occurs at least once (in fact, infinitely often). May 25 at 15:19

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. $$F[\alpha]\ =\ \{p(\alpha)\,:\,p\in F[X]\}\,.$$