Non polynomial terms and a rigorous theory of equations

Question

RECAP - WHAT I KNOW AND HOW I STUDIED IT

I started my logic/algebra course from studying propositional logic, then first order logic, then set theory, and then we started “building” natural, integer, rational and finally real numbers (as approximations by default, not by excess and default as Dedekind sections, see notes$^1$). Then, we started talking about monomials, polynomials and, finally, rational functions.

We study that a rational (or algebraic) function is defined as the equivalence class of a “rational term”, where a rational term is a couple $(p,q)$ of polynomials in a single variable with coefficients in a generic field $k$, named “polynomial fraction”, that satisfies certain rules. The equivalence relation is the one defined by $\frac{p}{q} \sim \frac{p’}{q’} \iff pq’=p’q$ and the quotient set is the field of rational functions.

For rational terms we can then define an “algebraic equation in one variable” as being a formula of type $s(x)=t(x)$ where $s$ and $t$ are rational terms with coefficients in a field $k$ and $x$ is the only variable. Then, we build the whole equation theory on this ”constraint” that everything that appears in this formula must be algebraic terms, and all the results are based on the concepts related to a rational term.

QUESTION

I have always took for granted that the ideas of “equivalent equations” and the “two principles of equations” translate to non algebraic equations, so equations in which appear anything that is not a rational term (such as n-th roots of the variable, trascendental functions with the variable as an argument etc.). But now I want to know: how can we study non algebraic equations in a rigorous way? Can you give me a preview? Is there a textbook or a source that you recommend?

• Note that I’m only interested in a rigorous way to define what a non-algebraic equation is and in seeing that you can still define the “equivalence” and “principles” of equationa. I’m not interested in general solving-methods, which, from what I know, don’t even exist

• As “two principle of equations” I refer to the result that operating on both sides of an equations, if you satisfy the correct hypothesises, yields an equivalent equation, as written here. I’m intersted particularly in this because it’s what gives us the ability to manipulate equations in the most basic way.

NOTES

$^1$ This method of constructing the rational numbers is called "ideals" by my textbook, so an ideal is defined as "a subset $\mathfrak a \subset \mathbb{Q}$ such that, if $x \in \mathfrak a$ and $y \leq x$ then $y \in \mathfrak a$". Then we define intervals, majorant and minorants, "closed ideals" etc. For example the "principal ideal of a $q \in \mathbb{Q}$ is $<q> = (-\infty, q]$. I see that it is a different definition of ideal than the wikipedia one, which seems more general.

Answer 1

There are two ways to define something like (say) $\sqrt{1-x^3}$ rigorously. The first approach you will find in most (abstract) algebra textbooks. It can be presented in superficially different but essentially equivalent ways. I can only sketch the ideas very briefly. We start with a field $K$ and an irreducible monic polynomial $f(z)\in K[z]$; we want to construct a field $E\supseteq K$ containing an element $\alpha$ such that $f(\alpha)=0$ in $E$. This is called adjoining a root.

The more concrete construction you will find, for example, in Emil Artin's Galois Theory, starting on p.26. We let $E$ be the set of all polynomials $g(z)$ of degree $<n$, where $n=\deg f$. $K\subseteq E$ by regarding elements of $K$ as polynomials of degree 0. Addition in $E$ is defined as usual, but we define multiplication (call it $\times$) in $E$ as follows: $$g\times h=gh\bmod f$$ where $gh$ is the usual polynomial multiplication, and $gh\bmod f$ is the remainder upon dividing by $f$. With this definition it can be shown that $E$ is a field, and that $f(z)=0$, where "$f(z)$" means the result of evaluating $f$ at $z$ using $\times$ for multiplication.

This follows the same pattern as used to define the field of $p$ elements where $p$ is a prime. This is $\mathbb{F}_p=\{0,\ldots,p-1\}$, with multiplication defined by $a\times b=ab\bmod p$.

In both cases, we can reframe this in a more abstract manner. Let's take the case of $\mathbb{F}_p$ first. Let $(p)$ be the set of all multiples of $p$, the so-called principal ideal generated by $p$. We say $a\equiv b\pmod{p}$ ($a,b\in\mathbb{Z}$) if $p|(a-b)$, i.e., if $a-b\in(p)$. $\mathbb{F}_p=\mathbb{Z}/(p)$ is defined as the set of equivalence classes. Addition and multiplication are defined by $[a]+[b]=[a+b]$ and $[a][b]=[ab]$. Each equivalence class $[a]$ has a unique representative in the set $\{0,\ldots,p-1\}$. The more concrete construction just deals always with these representatives.

It works the same way for adjoining a root: we say $g\equiv h\pmod{f}$ if $f|(g-h)$; $E=K[z]/(f)$, the set of equivalence classes; each equivalence class has a unique representative among the polynomials of degree $<\deg f$. It can be shown that if $\alpha$ is the equivalence class of the polynomial $z$, then $f(\alpha)=0$ in $E$.

This works fine for any field $K$. If $K$ happens to be $k(x)$, the field of rational functions in $x$, and (say) $f(z)=z^2-(1-x^3)$, then we have a field $E$ with an element $\alpha$ that we could denote $\sqrt{1-x^3}$, since $\alpha^2=1-x^3$.

You can iterate this procedure as many times as you like. Using either transfinite induction or Zorn's lemma, you can even define rigorously the algebraic closure of $k(x)$.

Besides Emil Artin's Galois Theory, most abstract algebra textbooks will cover this one way or another. This MSE question lists a bunch.

The other approach, not as commonly seen, is using so-called Puiseaux series. These are formal series where the exponents are allowed to be both negative and fractional. This works only in characteristic zero, but apparently there is a generalization (Hahn series) that works in all characteristics. I leave the details to the linked Wikipedia page.

You ask about transcendental functions, like say $\log x$. Here there are lots of approaches, not equivalent but related, depending on what you want to achieve. To mention just one: it's not hard to define the field of formal power series $K[[z]]$ over a field $K$.