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a/ws2022/rav/lecture/rav.tex +++ b/ws2022/rav/lecture/rav.tex @@ -33,5 +33,7 @@ Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.un \input{rav15.tex} \input{rav16.tex} \input{rav17.tex} +\input{rav18.tex} +\input{rav19.tex} \end{document} diff --git a/ws2022/rav/lecture/rav17.pdf b/ws2022/rav/lecture/rav17.pdf index 5c6ef95484a0f9ff1f87a02955059e58b4f9f7eb..91465316db3acb49ab387a8cf96e48315a8e3d8a 100644 GIT binary patch delta 12779 zcma)hbyQT*_b)R@H#3BUG)hT#cL~xB($d{Eq@>i5?h=q@Xrx1wmIhH2q(kXa9)7>S z_1^pMy>;(e=d<^nbzhi*#gD@%B173b!K|m+^1Tkl9=t?C> z9I{ZC8N5tdY~cs;q;MD$VNT}OT`i%e)$7so6-@kM(c;n0e{MJ)t;(M3Oj>;W_!QS5 z*lonJA+av;@A>YWsz8Tv-t_cpOk#-e&F|g)m4Y(rw9;9Vq*@onS41ouBTrPTH%qE= zrgOrH6(&9mXa1Nwp)pG+=t5k6*xQoKONvS!nnWP}{wQDAa;*#bbNG9=sj&~A%ZsG;u!0z_?GG54I-{1&nITS_zK04Vdp9By!3H&ZOK4xnHG@R9T4Fd4 zcfA><7$0yI#e|M3^~zH9D`vk@FnG?;p!a&XeqHO8=F$3-Y>!BdHYIdKa6}Ed7*F(k z>XH%7k0*IGWnsnEgo^XwY%VL!X0-1yEH~26=5Em$+8CEJIoH)ik{GjX;gT*V*O}uv 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degree are simpler from the real point of view: \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}}) + \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 @@ -214,4 +214,14 @@ Simple extensions of odd degree are simpler from the real point of view: \end{itemize} \end{bsp} +\begin{bem}[] + By \ref{thm:charac-real-closed}, if $k$ is a real-closed field, then the absolute galois + group of $k$ is + \[ + \text{Gal}(\overline{k} / k) = \text{Gal}(k[i] / k) \simeq \Z / 2 \Z + .\] The Artin-Schreier theorem shows that if $\overline{k} / k$ + is a non-trivial extension of \emph{finite} degree, + then $k$ is real-closed. +\end{bem} + \end{document} diff --git a/ws2022/rav/lecture/rav19.pdf b/ws2022/rav/lecture/rav19.pdf new 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zZ`+_D;c_D)^hc3nejSo<_>7dvQN@uI9Pgk= 0} \R((t ^{t/q})) + .\end{salign*} Then the subfield + $\overline{\R(t)}^{\widehat{\R(t)}}$, consisting of all those real + Puiseux series that are algebraic over $\R(t)$, is a real closure of $\R(t)$. + + The field of real Puiseux series itself is a real closure of the field $\R((t))$ + of real formal Laurent series. + \end{enumerate} +\end{bsp} + +Real-closed fields $L$ admit a canonical structure of ordered field, where $x \ge 0$ +in $L$, if and only if $x$ is a square. In particular, +if $k$ is a real field and $k^{r}$ is a real closure of $k$, then +$k$ inherits an ordering from $k^{r}$. However, different real closures may induce +different orderings on $k$, as the next example shows. + +\begin{bsp}[] + Let $k = \Q(t)$. This is a real field, since $\Q$ is real. Since $\pi$ + is transcendental over $\Q$, we can embed $\Q(t)$ in $\R$ by sending $t$ to $\pi$. + \[ + i_1\colon \Q(t) \xhookrightarrow{\simeq} \Q(\pi) \subseteq \R + .\] Since $\R$ is real-closed, the relative algebraic closure + $i_1(\Q(t))^{\R}$ is a real closure of $i_1(\Q(t))$. + + We can also embed $\Q(t)$ in the field $\widehat{\R(t)}$ of real Puiseux series via + a homomorphism $i_2$ and then + $\overline{i_2(\Q(t))}^{\widehat{\R(t)}}$ is a real closure of $i_2(\Q(t))$. + However, the ordering on $\overline{i_1(\Q(t))}^{\R}$ + is Archimedean, because it is a subfield of $\R$, + while the ordering on $\overline{i_2(\Q(t))}^{\widehat{\R(t)}}$ + is not Archimedean (it contains infinitesimal elements, such as $t$ for instance). + + The fields $\overline{i_1(\Q(t))}^{\R}$ + and $\overline{i_2(\Q(t))}^{\widehat{\R(t)}}$ cannot be isomorphic as fields. + Indeed, when two real-closed fields $L_1, L_2$ are isomorphic as fields, + then they are isomorphic as ordered fields, since positivity on a real + closed field is defined by the condition of being a square, which is preserved + under isomorphisms of fields. +\end{bsp} + +The next result will be proved later on. + +\begin{lemma}[] + Let $(k, \le )$ be an ordered field and $P \in k[t]$ be an irreducible polynomial. + Let $L_1, L_2$ be real-closed extensions of $k$ that are compatible with the ordering of $k$. + Then $P$ has the same number of roots in $L_1$ as in $L_2$. +\end{lemma} + +\end{document}