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--git a/ws2022/rav/lecture/rav16.tex b/ws2022/rav/lecture/rav16.tex new file mode 100644 index 0000000..5da1e1e --- /dev/null +++ b/ws2022/rav/lecture/rav16.tex @@ -0,0 +1,284 @@ +\documentclass{lecture} + +\begin{document} + +\chapter{Real algebra} + +\section{Ordered fields and real fields} + +\begin{definition}[] + An \emph{ordered field} is a pair $(k, \le)$ consisting of a field $k$ and + an order relation $\le$ such that + \begin{enumerate}[(i)] + \item $\le $ is a total order: if $x, y \in k$, then $x \le y$ or $y \le x$. + \item $\le $ is compatible with addition in $k$: + if $x, y, z \in k$, then $x \le y$ implies $x + z \le y + z$. + \item $\le $ is compatible with multiplication in $k$: + if $x, y\in k$, then $0 \le x$ and $0 \le y$ implies $0 \le xy$. + \end{enumerate} + A morphism between two ordered fields $(k, \le)$ and $(L, \le)$ is a field homomorphism + $\varphi\colon k \to L$ such that $x \le y$ in $k$ implies $\varphi(x) \le \varphi(y)$ in $L$. +\end{definition} + +\begin{bsp}[] + \begin{enumerate}[(1)] + \item The fields $\Q$ and $\R$, equipped with their usual orderings, are + ordered fields. + \item The field $\mathbb{C}$ can be equipped with a total ordering + (the ,,lexicographic order``) but not with a structure of + ordered field. + \item The field $\R(t)$ of rational fractions with coefficients in $\R$, can + be equipped with a structure of ordered field in multiple ways: + + Fix an $x \in \R$ and, for all polynomial $P \in \R[t]$, use + Taylor expansion at $x$ to write + \[ + P(t) = a_p (t - x)^{p} + \text{higher order terms} + .\] + with $a_p \neq 0$, then define + $P(t) >_{x^{+}} 0$ if $a_p > 0$, i.e. if the function + $t \mapsto P(t)$ is positive on a small interval $(x, x + \epsilon)$. Set also + $\frac{P(t)}{Q(t)} >_{x^{+}} 0$ if $P(t)Q(t) >_{x^{+}} 0$, + and define $f \le_{x^{+}} g$ in $\R(t)$ if either $f = g$ or + $g - f >_{x^{+}} 0$. + Equivalently $f \le_{x^{+}} g$ in $\R(t)$ if + either $f = g$ or $g - f$ is positively-valued on $(x, x + \epsilon)$ for $\epsilon > 0$ small + enough. + + It is clear that this is a total ordering on $\R(t)$, and that this ordering is compatible + with addition and multiplication in the sense of the definition of an ordered field. + Moreover, the substitution homomorphism + $h(t) \mapsto h(t - x)$ induces an isomorphism of ordered fields + $(\R(t), \le_{0^{+}}) \xlongrightarrow{\simeq} (\R(t), \le_{x^{+}})$, + since a function $t \mapsto h(t - x)$ is positively-valued + on $(x, x + \epsilon)$ if and only if the function $t \mapsto h(t)$ is positively valued + on $(0, \epsilon)$. + + Note that we can also define orderings on $\R(t)$ by setting $f \le_{x^{-}} g$ + if either $f = g$ or $g - f$ is positively-valued + on $(x-\epsilon, x)$, for $\epsilon > 0$ small enough. + The substitution homomorphism $h(t) \mapsto h(-t)$ induces an isomorphism + of ordered fields + $(\R(t), \le_{0^{-}}) \xlongrightarrow{\simeq} (\R(t), \le_{0^{+}})$. + \end{enumerate} +\end{bsp} + +\begin{bem}[] + The ordered field $(\R(t), \le_{0^{+}})$ + is non-Archimedean: the element $t$ is + \emph{infinitely small with respect to any real $\delta > 0$} in the sense that for all $n \in \N$, + $nt < \delta$ (indeed $t \mapsto n t - \delta$ is negatively-valued + on $(0, \epsilon)$ for $\epsilon > 0$ small enough). Equivalently, $\frac{1}{t}$ + is infinitely large with respect to $ 0 < \delta \in \R$ in the sense that + $\frac{1}{t} > n \delta$ for all $n \in \N$. +\end{bem} + +\begin{satz}[] + Let $(k, \le)$ be an ordered field and $x, y, z \in k$. Then the following properties hold: + \begin{enumerate}[(a)] + \item $x \ge 0$ or $- x \ge 0$. + \item $-1 < 0$ and $1 > 0$. + \item $k$ is of characteristic $0$. + \item if $x < y$ and $z > 0$, then $x z < y z$. + \item if $x < y$ and $z < 0$, then $x z > y z$. + \item $x y \ge 0$ if and only if $x$ and $y$ have the same sign. + \item $x^2 \ge 0$ and, if $x \neq 0$, then $x$ and $\frac{1}{x}$ have the same sign. + \item if $0 < x \le y$, then $0 < \frac{1}{y} \le \frac{1}{x}$. + \end{enumerate} + \label{satz:ordered-field-basics} +\end{satz} + +\begin{proof} + Elementary verifications. +\end{proof} + +It turns out that it is possible to characterise ordered fields without explicitly mentioning +the order relation, using cones of positive elements. + +\begin{definition} + Let $k$ be a field. A \emph{cone} in $k$ is a subset $P \subseteq k$ such that + for all $x, y \in P$ and $z \in k$: + \begin{enumerate}[(i)] + \item $x + y \in P$ + \item $xy \in P$ + \item $z^2 \in P$ + \end{enumerate} + A cone $P \subseteq k$ is called a \emph{positive cone} if, additionally, one has: + \begin{enumerate}[(i)] + \setcounter{enumi}{3} + \item $-1 \not\in P$ + \end{enumerate} +\end{definition} + +\begin{satz}[] + Let $k$ be a field. Assume that there exists a positive cone $P \subseteq k$. Then: + \begin{enumerate}[(i)] + \item $0 \in P$ and $1 \in P$. + \item $k$ is of characteristic $0$. + \item $P \cap (-P) = \{0\}$ + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(i)] + \item $0 = 0^2 \in P$ and $1 = 1^2 \in P$ by axiom (iii). + \item Since $1 \in P$, by induction and axiom (i), + $n \cdot 1 = \underbrace{1 + \ldots + 1}_{n \text{ times}} \in P$ for all $n \in \N$. + Assume that there exists $n \in \N$, such that $n \cdot 1 = 0$ in $k$. + Since $1 \neq 0$ in $k$, it follows $n \ge 2$ so, + \[ + -1 = 0 - 1 = n \cdot 1 - 1 = (n - 1) \cdot 1 \in P + ,\] which contradicts axiom (iv). + \item Assume that there exists $x \in P \cap (-P) \setminus \{0\}$. In particular + $x \neq 0$ and $-x \in P$. So + $- x^2 = (-x) x \in P$ by axiom (ii) and $\frac{1}{x^2} = \left( \frac{1}{x} \right)^2 \in P$ + by axiom (iii). Again by axiom (ii) + \[ + -1 = \frac{1}{x^2} (-x^2) \in P + \] which contradicts axiom (iv). + \end{enumerate} +\end{proof} + +Given a positive cone $P$ in a field $k$, let us set $P^{+} = P \setminus \{0\} $ +and $P^{-} = (-P) \setminus \{0\} = - P^{+}$. Then we have a disjoint union +\[ +P^{-} \sqcup \{0\} \sqcup P^{+} \subseteq k +.\] +Note that $P^{+}$ satisfies axioms (i) and (ii) of the definition of a cone, as well +as the property that $x \in k \setminus \{0\} \implies x^2 \in P^{+}$. + +We now prove that positive curves can be enlarged, that the resulting notion of +maximal positive cone satisfies $P \cup (-P) = k$, and that +this defines a structure of ordered field on $k$ by setting $x \le y$ if and only if $y - x \in P$. + +\begin{lemma} + Assume that $P$ is a positive cone in a field $k$. If $a \in k \setminus P \cup (-P)$, then the set + \[ + P[a] \coloneqq \{ x + a y \in k \colon x, y \in P\} + \] + is a positive cone in $k$, satisfying $P \subsetneq P[a]$. + \label{lemma:positive-cone-extend-by-one-element} +\end{lemma} + +\begin{proof} + Let $x, y, x', y' \in P$. Then + \[ + (x + ay) + (x' + a y') = x + x' + a(y + y') \in P[a] + \] and + \[ + (x+ay)(x' + ay') = x x' + a^2 y y' + a (x y' + x' y) \in P[a] + .\] Moreover $z^2 \in P \subseteq P[a]$ for all $z \in k$. + + Now assume $-1 = x + a y$ for some $x, y \in P$. + If $y = 0$, then $-1 = x \in P$ which is a contradiction. Thus $y \neq 0$ and + \[ + - a = \frac{1 + x}{y} = \left( \frac{1}{y} \right)^2 y (1+x) \in P + ,\] which contradicts the assumption on $a$. Finally, we have $P \subseteq P[a]$ and, + if $P[a] \subseteq P$ then $a \in P$, again contradicting the assumption on $a$. So + $P \subsetneq P[a]$. +\end{proof} + +\begin{satz} + Let $\mathcal{P}$ be the set of positive cones of a field $k$ ordered + by inclusion. If $\mathcal{P} \neq \emptyset$, then + $\mathcal{P}$ admits a maximal element and such an element $P$ satisfies + $P \cup (-P) = k$. + \label{satz:existence-maximal-positive-cones} +\end{satz} + +\begin{proof} + To obtain a maximal element of $\mathcal{P}$, + by Zorn's lemma, it suffices to show, that every + chain $(P_i)_{i \in I}$ in $\mathcal{P}$ has an upper bound. We set + \[ + P = \bigcup_{i \in I} P_i \subseteq k + .\] One verifies immediately that $P$ is a positive cone and an upper bound of the chain $(P_i)_{i \in I}$. + + Let $P$ be such a maximal element. If there exists $a \in k \setminus P \cup (-P)$, then by + \ref{lemma:positive-cone-extend-by-one-element} $P \subsetneq P[a]$ contradicts the maximality of $P$. Thus + $P \cup (-P) = k$. +\end{proof} + +\begin{satz} + Let $k$ be a field and denote by + \[ + \Sigma k^{[2]} \coloneqq + \left\{ y \in k \mid \exists (a_x)_{x \in k} \in \{0, 1\}^{(k)}, y = \sum_{x \in k} a_x x^2 \right\} + \] + the set of sums of squares in $k$. Then + $\Sigma k^{[2]}$ is a cone and $-1 \not\in \Sigma k^{[2]}$ if and only if + for all $x_1, \ldots, x_n \in k$: + \[ + x_1^2 + \ldots + x_n^2 = 0 \implies x_1 = \ldots = x_n = 0 + .\] + \label{satz:sums-of-squares-cone} +\end{satz} + +\begin{proof} + One verifies immediately that $\Sigma k^{[2]}$ is a cone in $k$. If + $-1 \in \Sigma k^{[2]}$, then + $-1 = x_1^2 + \ldots + x_n^2$ for some $x_i \in k$. Thus + \[ + 0 = \sum_{i=1}^{n} x_i^2 + 1 + \] but $1 = 1^2$ and $1 \neq 0$. Conversely let + $0 = \sum_{i=1}^{n} x_i^2$ with $x_1 \neq 0$. Then + \[ + -1 = \frac{1}{x_1^2} \sum_{i=2}^{n} x_i^2 = + \sum_{i=2}^{n} \left(\frac{x_i}{x_1}\right)^2 + \in \Sigma k^{[2]} + .\] +\end{proof} + +\begin{definition} + A field $k$ is called a \emph{real field} if $-1 \not\in \Sigma k^{[2]}$, or equivalently + if $\sum_{k=1}^{n} x_i^2 = 0$ in $k$ implies $x_k = 0$ for all $k$. +\end{definition} + +\begin{korollar} + Let $k$ be a field. $k$ is real if and only if $k$ contains + a positive cone. +\end{korollar} + +\begin{proof} + $(\Rightarrow)$: By \ref{satz:sums-of-squares-cone} $\Sigma k^{[2]}$ is a positive + cone. + $(\Leftarrow)$: Let $P$ be a positive cone. Since + $P$ is closed under addition and for all $z \in k\colon z^2 \in P$, + $\Sigma k^{[2]} \subset P$. Since $P$ is positive, $-1 \not\in \Sigma k^{[2]}$. +\end{proof} + +\begin{satz} + Let $(k, \le)$ be an ordered field. Then the set + \[ + P \coloneqq \{ x \in k \mid x \ge 0\} + \] is a maximal positive cone in $k$. In particular, + $k$ is a real field. Conversely, if $k$ is a real field and $P$ is a maximal + positive cone in $k$, then the relation $x \le_P y$ if $y - x \in P$ is an order + relation and $(k, \le_P)$ is an ordered field. +\end{satz} + +\begin{proof} + $(\Rightarrow)$: + Let $(k, \le )$ be an ordered field. Then by + definition and \ref{satz:ordered-field-basics}, $P$ is a maximal positive cone. + + $(\Leftarrow)$: Let $P$ be a maximal positive cone in $k$. Since + $0 \in P$, we have $x \le_P x$. Suppose that $x \le_P y$ and $y \le_P x$. Then + $y - x \in P \cap (-P) = \{0\} $, so $x = y$. Moreover, if $x \le_P y$ + and $y \le_P z$, then $z - x = (z - y) + (y - x) \in P$. Thus $x \le_P z$, hence + $\le_P$ is an order relation. Moreover, it is a total order, because if + $x, y \in k$, then $y - x \in k = P \cup (-P)$, so either $x \le_P y$ or $y \le_P x$. + + Finally, + this total order on $k$ is compatible with addition and multiplication because + $x \le_P y$ and $z \in k$ implies $(y + z) - (x + z) = y - x \in P$, so + $x + z \le_P y + z$, and $x \ge_P 0$, $y \ge_P 0$ means that $x \in P$ and $y \in P$, so $xy \in P$, + hence $xy \ge_P 0$. +\end{proof} + +\begin{korollar} + Let $k$ be a field. Then $k$ admits a structure of ordered field + if and only if $k$ is real. +\end{korollar} + +\end{document}