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z)Tjc)P-S{h5Rx+r+F{NuCVrNrW<|sWRox+0))ac@YhOP^1C=IE!hl#CAUIqZY)+jN*za0dwnajz2I7L6B z20Ssp$3t%n&3{d}f-RN%VKN<5ff5D}RUjd>#4k*iRH^(`|IC$zOQuhViAWfILsV#; zRY5SAnLxP;mBSI_fnMB>hRKxBt!iZzTp^yYitg4bc|u(?OkAM`u&?Vjn%@Suo$weS zFdSSkZf)P26a3V{#U1XlQqvW|Zmf|uKEBFbjmA1>eplE?1~Rt5iYby*#9$9VY1}+0 zhm6VB6k!}40k%)Os}wc58&7dC{aThfaM~k)=PZGR*X}=1oQjf?^=kX(#5J$zU23Rj zYqkj6JTrL8Jy7MPx-^|H$Y{*3!$ixJvDmn}%k|e_5+Wqn<2xblENlLv%^Fx#%}cVm zd&Gylm)IZs^2;&buJg1|p=73q*ypbK(ljRmjWQg^(1k<0c+ys^>X&i5b2jsHl79sd zh^d8)69qf~h!sL)JObnk?}hem@lKXQunO|i|2FVpiXUOXy?Zh1y50jRpw+n|p{3=% z(}cuIJu>)_hhFQ2tX_xgQa_pv+Cl4Vk-h zG7A7AO}zh 0$ for all $i$, then $q$ and $-q$ are anisotropic on $k^{n}$. + \end{itemize} +\end{bsp} + +\begin{definition} + Let $k$ be a field and $L$ an extension of $k$. A quadratic form + $q\colon k^{n} \to k$ induces a quadratic form $q_L \colon L^{n} \to L$. The form + $q$ is called \emph{anisotropic over $L$} if $q_L$ is anisotropic. +\end{definition} + +It can be checked that, on an ordered field $(k, \le )$, a quadratic form +$q$ is anisotropic if and only if it is non-degenerate and of constant sign. The interest +of this notion for us is given by the following result. + +\begin{theorem} + \label{thm:charac-orderable-extension} + Let $(k, \le )$ be an ordered field and $L$ be an extension of $k$. Then the following + conditions are equivalent: + \begin{enumerate}[(i)] + \item The extension $L / k$ is orderable. + \item For all $n \ge 1$ and all $a = (a_1, \ldots, a_n) \in k^{n}$ such that + $a_i > 0$ for all $i$, the quadratic form + $q(x_1, \ldots, x_n) = a_1x_1^2 + \ldots + a_n x_n^2$ is anisotropic over $L$ + (i.e. all positive definite quadratic forms on $k$ are anisotropic over $L$). + \end{enumerate} +\end{theorem} + +\begin{proof} + (i)$\Rightarrow$(ii): Assume that there is an ordering of $L$ that extends + the ordering of $k$ and let $n \ge 1$. Let $a = (a_1, \ldots, a_n) \in k^{n}$ + with $a_i > 0$ for all $i$. Then $a_i > 0$ still holds in $L$. Since + squares are non-negative for all orderings, the sum + $a_1 x_1^2 + \ldots + a_n x_n^2$ is a sum of positive terms in $L$. Therefore + it can only be $0$, if all of its terms are $0$. Since $a_i \neq 0$, it follows + $x_i = 0$ for all $i$. + + (ii)$\Rightarrow$(i): Define + \[ + P = \bigcup_{n \ge 1} \left\{ \sum_{i=1}^{n} a_i x_i^2 \colon a_i \in k, a_i > 0, x_i \in L \right\} + .\] The set $P$ is stable by sum and product and contains all squares of $L$, + so it is a cone in $L$. Suppose $-1 \in P$. Then there exists + $n \ge 1$ and $a = (a_1, \ldots, a_n) \in k^{n}$ with $a_i > 0$ + and $x = (x_1, \ldots, x_n) \in L^{n}$ such that + $-1 = \sum_{i=1}^{n} a_i x_i^2$. So + \[ + a_1 x_1^2 + \ldots + a_n x_n^2 + 1 = 0 + ,\] meaning that the quadratic form $a_{1} x_1^2 + \ldots + a_n x_n^2 + x_{n+1}^2$ + is isotropic on $L^{n+1}$, contradicting (ii). + Thus $P$ is a positive cone containing all positive elements of $k$. By + embedding $P$ in a maximal positive cone, the claim follows. +\end{proof} + +\begin{satz}[] + Let $(k, \le )$ be an ordered field and let $c > 0$ be a positive element in $k$. + Then $k[\sqrt{c}]$ is an orderable extension of $k$. +\end{satz} + +\begin{proof} + If $c$ is a square in $k$, there is nothing to prove. Otherwise, $k[\sqrt{c}]$ is + indeed a field. Let $n \ge 1$ and let $a = (a_1, \ldots, a_n) \in k^{n}$ + with $a_i > 0$ for all $i$. Assume that $x = (x_1, \ldots, x_n) \in k[\sqrt{c}]^{n}$ + satisfies + \[ + a_1 x_1^2 + \ldots + a_n x_n^2 = 0 + .\] Since $x_i = u_i + v_i \sqrt{c} $ for some $u_i, v_i \in k$, we can rewrite + this equation as + \[ + \sum_{i=1}^{n} a_i (u_i^2 + c v_i^2) + 2 \sum_{i=1}^{n} u_i v_i \sqrt{c} = 0 + .\] + Since $1$ and $\sqrt{c}$ are linearly independent over $k$, we get + $\sum_{i=1}^{n} a_i (u_i^2 + c v_i^2) = 0$, hence + $u_i = v_i = 0$ for all $i$, since all terms in the previous sum are non-negative. + So $x_i = 0$ for all $i$ and (ii) of \ref{thm:charac-orderable-extension} + is satisfied. +\end{proof} + +\begin{satz} + Let $(k, \le )$ be an ordered field and let $P \in k[t]$ be an irreducible + polynomial of odd degree. Then the field $L \coloneqq k[t]/ (P)$ is an orderable + extension of $k$. +\end{satz} + +\begin{proof} + Denote by $d$ the degree of $P$ and proceed by induction on $d \ge 1$. If $d = 1$, then + $L = k$. Now assume $d \ge 2$. Let $n \ge 1$ and $a_1, \ldots, a_n \in k$ with $a_i > 0$. + Denote by $q_L$ the quadratic form + \[ + q_L(x_1, \ldots, x_n) = a_1 x_1^2 + \ldots + a_n x_n^2 + \] on $L^{n}$. If $q_L$ is isotropic over $L$, then there exist + polynomials $g_1, \ldots, g_n \in k[t]$ with $\text{deg}(g_i) < d$ + and $h \in k[t]$ such that + \begin{equation} + q_L(g_1, \ldots, g_n) = h P + \label{eq:quad-form} + \end{equation} + Let $g$ be the greatest common divisor of $g_1, \ldots, g_n$. Since $q_L$ is + homogeneous of degree $2$, + $g^2$ divides $q_L(g_1, \ldots, g_n)$. Since $P$ is irreducible, $g$ divides $h$. + We may thus assume that $g = 1$. The leading coefficients of the terms on + the left hand side of (\ref{eq:quad-form}) are non-negative, thus + the sum has even degree $< 2d$. Since the degree of $P$ is odd, + $h$ must be of odd degree $< d$. Therefore, $h$ has an irreducible factor + $h_1 \in k[t]$ of odd degree. Let $\alpha$ be a root of $h_1$. By evaluating + (\ref{eq:quad-form}) at $\alpha$, we get + \[ + q_{k[\alpha]}(g_1(\alpha), \ldots, g_n(\alpha)) = 0 + \] in $k[\alpha]$. Since the $gcd(g_1, \ldots, g_n) = 1$ and $k[t]$ is a principal ideal + domain, there exist $h_1, \ldots, h_n \in k[t]$ such that + \[ + h_1 g_1 + \ldots + h_n g_n = 1 + .\] In particular + \[ + h_1(\alpha) g_1(\alpha) + \ldots + h_n(\alpha) g_n(\alpha) = 1 + ,\] so not all $g_i(\alpha)$ are $0$ in $k[\alpha]$. Thus $q_{k[\alpha]}$ + is isotropic over $k[\alpha] = k[t] / (h_1)$ contradicting the induction hypothesis. +\end{proof} + +\end{document}