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  • NcDJtF+{pby;Y6-%Kvm?n)41sD17vQeLX(0L}h-nA0ToU&l#D4mcVTSimK@kHNx9 zkQW0MrGv#GU`Z%gngo{dl*{>Tg#%a_4_0yFtLs31J}77aYrcZDzrZ?9tuRkD=}mly zuN3P!rVYO28`xwBHW!1Uk6?=pC}zE_H^8<6u)PtK^nso3s-8ayAc0csGD2dv1=wRx z{(;gaur~+nGXZ5>QtkvQmVy0c;J{~a@HaU09UQI)M|eGzkH9f@bzB#mNCYPXz^N{9 z`U5z_ZD;?2DpPRI3RDYF!*#W6qn0zMYXn(0=TUM?y!yaW1!Hm{imXEDb4#YG(t-6sVbT+WxB44=193S zSw(ZDtZGwHq?DauDvFZwi>ZpDr8eNciejWT3>|2uW?&TlB-$99SF=E;+W1^xy%9{h z4D4|;nqzWN%_> b3UhQ}a&&ldWo8O7G&wgiFbX9lMNdWwY`nb- literal 0 HcmV?d00001 diff --git a/ws2022/rav/lecture/rav17.tex b/ws2022/rav/lecture/rav17.tex new file mode 100644 index 0000000..6106523 --- /dev/null +++ b/ws2022/rav/lecture/rav17.tex @@ -0,0 +1,217 @@ +\documentclass{lecture} + +\begin{document} + +\section{Real-closed fields} + +In this section we study real algebraic extensions of real fields. + +\begin{lemma} + Let $k$ be a real field and $x \in k \setminus \{0\} $. Then + $x$ and $-x$ cannot be both sums of squares in $k$. + \label{lemma:real-field-only-one-is-square} +\end{lemma} + +\begin{proof} + If $x \in \Sigma k^{[2]}$ and $-x \in \Sigma k^{[2]}$, then + \[ + 1 = \frac{1}{x^2} (-x) x \in \Sigma k^{[2]} + \] contradicting that $k$ is real. +\end{proof} + +\begin{satz} + Let $k$ be a real field and $a \in k$ such that $a$ is not a square in $k$. + Then the field + \[ + k(\sqrt{a}) = k[t] / (t^2 - a) + \] is real if and only if $-a \not\in \Sigma k^{[2]}$. + In particular, if $\Sigma k^{[2]} \cup (- \Sigma k^{[2]}) \neq k$, + then $k$ admits real quadratic extensions. + \label{satz:quadratic-extensions-of-real-field} +\end{satz} + +\begin{proof} + Since $a$ is not a square in $k$, $t^2 - a$ is irreducible in $k[t]$, so + $k[t] / (t^2 -a)$ is indeed a field. Denote by $\sqrt{a} $ the class of + $t$ in the quotient. + + ($\Rightarrow$): $a$ is a square in $k(\sqrt{a})$, thus by + \ref{lemma:real-field-only-one-is-square} we have $-a \not\in \Sigma k(\sqrt{a})^2$. + But $\Sigma k^{[2]} \subseteq \Sigma k(\sqrt{a})^{[2]}$, thus + $-a \not\in \Sigma k(\sqrt{a})^{[2]}$. + + ($\Leftarrow$): + $-1 \in \Sigma k(\sqrt{a})^{[2]}$ + if and only if there exist $x_i, y_i \in k$, such that + \[ + -1 = \sum_{i=1}^{n} (x_i + y_i \sqrt{a})^2 + = \sum_{i=1}^{n} (x_i^2 + a y_i^2) + 2 \sqrt{a} \sum_{i=1}^{n} x_i y_i + .\] + Since $(1, \sqrt{a})$ is a basis of the $k$-vector space $k(\sqrt{a})$, the previous equality + implies + \begin{salign*} + -1 &= \sum_{i=1}^{n} x_i^2 + a \sum_{i=1}^{n} y_i^2 + .\end{salign*} + Since $-1 \not\in \Sigma k^{[2]}$, $\sum_{i=1}^{n} y_i^2 \neq 0$, this + implies + \[ + -a = \frac{1 + \sum_{i=1}^{n} x_i^2}{\sum_{i=1}^{n} y_i^2} + = \frac{\left( \sum_{i=1}^{n} y_i^2 \right)\left( 1 + \sum_{i=1}^{n} x_i^2 \right) } + {\left( \sum_{i=1}^{n} y_i^2 \right)^2} + \in \Sigma k^{[2]} + .\] +\end{proof} + +Simple extensions of odd degree are simpler from the real point of view: + +\begin{satz} + Let $k$ be a real field and $P \in k[t]$ be an irreducible polynomial of odd degree. + Then the field $k[t]/(P)$ is real. + \label{satz:odd-real-extension} +\end{satz} + +\begin{proof} + Denote by $n$ the degree of $P$. We proceed by induction on $n \ge 1$. + If $n = 1$, then $k[t]/(P) \simeq k$ is real. Since $n$ is odd, we + may now assume $n \ge 3$. + Let $L \coloneqq k[t]/(P)$. Suppose $L$ is not real. Then there exist + polynomials $g_i \in k[t]$, of degree at most $n-1$, such that + $-1 = \sum_{i=1}^{m} g_i^2$ + in $L = k[t]/(P)$. Since $k \subseteq L$ and $k$ is real, at least + one of the $g_i$ is non-constant. + By definition of $L$, there exists $Q \in k[t] \setminus \{0\}$ + such that + \begin{equation} + -1 = \sum_{i=1}^{m} g_i^2 + P Q + \label{eq:gi-sq+pq} + \end{equation} + in $k[t]$. Since $k$ is real, in $\sum_{i=1}^{m} g_i^2$ no cancellations + of the terms of highest degree can occur. Thus + $\sum_{i=1}^{m} g_i^2$ is of positive, even degree at most $2n-2$. By + \ref{eq:gi-sq+pq}, it follows that $Q$ is of odd degree at most $n-2$. + In particular, $Q$ has at least one irreducible factor $Q_1$ of odd degree at most + $n-2$. Since $n \ge 3$, $n-2 \ge 1$. By induction, + $M \coloneqq k[t]/(Q_1)$ is real. But \ref{eq:gi-sq+pq} implies + \[ + -1 = \sum_{i=1}^{m} g_i^2 + \] in $M = k[t] / (Q_1)$ contradicting the fact that $M$ is real. +\end{proof} + +\begin{definition} + A \emph{real-closed} field is a real field that + has no proper real algebraic extensions. +\end{definition} + +\begin{theorem} + Let $k$ be a field. Then the following conditions are equivalent: + \begin{enumerate}[(i)] + \item $k$ is real-closed. + \item $k$ is real and for all $a \in k$, either $a$ or $-a$ + is a square in $k$ and + every polynomial of odd degree in $k[t]$ has a + root in $k$. + \item the $k$-algebra + \[ + k[i] \coloneqq k[t] / (t^2+1) + \] is algebraically closed. + \end{enumerate} + \label{thm:charac-real-closed}. +\end{theorem} + +\begin{proof} + (i)$\Rightarrow$(ii): Let $a \in k$ such that neither $a$ nor $-a$ is a square in $k$. Then + by \ref{satz:quadratic-extensions-of-real-field} and (i), $\pm a \in \Sigma k^{[2]}$ + contradicting + \ref{lemma:real-field-only-one-is-square}. Let $P \in k[t]$ be a polynomial + of odd degree. $P$ has at least one irreducible factor $P_1$ of odd degree. + By \ref{satz:odd-real-extension}, $k[t]/(P_1)$ is a real extension of $k$. + Since $k$ is real-closed, $P_1$ must be of degree $1$ and thus $P$ has a root in $k$. + + (ii)$\Rightarrow$(iii): Since $-1$ is not a square in $k$, the polynomial + $t^2 + 1$ is irreducible over $k$. Thus $L \coloneqq k[t]/(t^2 + 1)$ is a field. Denote + by $i$ the image of $t$ in $L$ and for $x = a + ib \in L = k[i]$, denote + by $\overline{x} = a - ib$. This extends to a ring homomorphism $L[t] \to L[t]$. Let + $P \in L[t]$ be non-constant. It remains to show, that $P$ has a root in $L$. We + first reduce to the case $P \in k[t]$. + + Assume every non-constant polynomial in $k[t]$ has a root in $L$. Let $P \in L[t]$. Then + $P \overline{P} \in k[t]$ has a root $x \in L$, thus either $P(x) = 0$ + or $\overline{P}(x) = 0$. In the first case, we are done. + In the second case, we have $P(\overline{x}) = \overline{\overline{P}(x)} + = \overline{0} = 0$, so $\overline{x}$ is a root of $P$ in $L$. + + Thus we may assume $P \in k[t]$. Write $d = \text{deg}(P) = 2^{m} n$ with $2 \nmid n$. We + proceed by induction on $m$. If $m = 0$, the result is true by (ii). Now assume $m > 0$. + Fix an algebraic closure $\overline{k}$ of $k$. Since $k$ is real, it is of characteristic + $0$, thus $k$ is perfect and $\overline{k} / k$ is galois. + Let $y_1, \ldots, y_d$ be the roots + of $P$ in $\overline{k}$. Consider for all $r \in \Z$: + \[ + F_r \coloneqq \prod_{1 \le p < q \le d}^{} + \left( t - (y_p + y_q) - r y_p y_q) \right) \in \overline{k}[t] + .\] This polynomial with coefficients in $\overline{k}$ is invariant + under permutation of $y_1, \ldots, y_d$. Thus its coefficients + lie in $\overline{k}^{\text{Gal}(\overline{k} / k)} = k$. Moreover + \[ + \text{deg}(F_r) = \binom{d}{2} = \frac{d(d-1)}{2} = 2^{m-1} n (2^{m} -1) + .\] with $n (2^{m} -1)$ odd. So the induction hypothesis applies and, + for all $r \in \Z$, there is a pair $p < q$ in $\{1, \ldots, d\} $ + such that $(y_p + y_q) + r y_p y_q \in L$. Since $\Z$ is infinite, + we can find a pair $p < q$ in $\{1, \ldots, d\} $ for which + there exists a pair $r \neq r'$ such that + \begin{salign*} + &(y_p + y_q) + r y_p y_q \in L \\ + \text{and } & (y_p + y_q) + r' y_p y_q \in L + .\end{salign*} + By solving the system, we get $y_p + y_q \in L$ and $y_p y_q \in L$. But $y_p, y_q$ + are roots of the quadratic polynomial + \[ + t^2 - (y_p + y_q)t + y_p y_q \in L[t] + \] and since $i^2 = -1$, the roots of this polynomial lie in $L = k[i]$, by (ii) and the + usual formulas + \[ + t_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a} + .\] So $P$ indeed has a root in $k[i]$, which finishes the induction. + + (iii)$\Rightarrow$(i): Denote again by $i$ the image in the algebraically closed field + $k[t]/(t^2 + 1)$. We first show that $k^{[2]} = \Sigma k^{[2]}$. Let $a, b \in k$. Then + $a + ib = (c + id)^2$ in $k[i]$ for some $c, d \in k$. Thus + \[ + a^2 + b^2 = (a+ib)(a-ib) = (c+id)^2(c-id)^2 = (c^2 + d^2)^2 + .\] By induction the claim follows. Since $t^2 + 1$ is irreducible, + $-1 \not\in k^{[2]} = \Sigma k^{[2]}$ and $k$ is real. + + Let $L$ be a real algebraic extension of $k$. Since $k[i]$ is algebraically closed and contains + $k$, there exists a $k$-homomorphism $L \xhookrightarrow{} k[i]$. Since + $[ k[i] : k ] = 2$, either $L = k$ or $L = k[i]$, but $k[i]$ is not real, since $i^2 = -1$ + in $k[i]$. So $L = k$ and $k$ is real-closed. +\end{proof} + +\begin{korollar} + A real-closed field $k$ admits a canonical structure of ordered field, in + which the cone of positive elements is exactly $k^{[2]}$, the set of squares in $k$. +\end{korollar} + +\begin{proof} + This was proven in the implication (i)$\Rightarrow$(ii) of \ref{thm:charac-real-closed}. +\end{proof} + +\begin{bsp}[] + \begin{itemize} + \item $\R$ is a real-closed field, because $\R[i] = \mathbb{C}$ is algebraically closed. + \item The field of real Puiseux series + \begin{salign*} + \widehat{\R(t)} \coloneqq \bigcup_{q > 0} \R(t ^{\frac{1}{q}}) + = \left\{ + \sum_{n=m}^{\infty} a_n t ^{\frac{n}{q}} \colon + m \in \Z, q \in \N \setminus \{0\}, a_n \in \R + \right\} + \end{salign*} + is a real closed field because + $\widehat{\R(t)}[i] = \widehat{\R[i][t]} = \widehat{\mathbb{C}[t]}$ is the field + of complex Puiseux series, which is algebraically closed by the + Newton-Puiseux theorem. + \end{itemize} +\end{bsp} + +\end{document}