|
Autor |
Galois-Gruppe bestimmen |
|
LukasNiessen
Aktiv  Dabei seit: 30.09.2019 Mitteilungen: 148
Herkunft: Nordrhein-Westfalen, Bonn, Weststadt
 | \(\begingroup\)\(\newcommand{\defi}{\overset{\mathscr{D}\mathscr{e}\mathscr{f}.}{=\!=}}
\newcommand{\defeq}{\overset{\mathscr{D}\mathscr{e}\mathscr{f}.}{=\!=}}
\newcommand{\IN}{\mathbb{N}}
\newcommand{\IZ}{\mathbb{Z}}
\newcommand{\IQ}{\mathbb{Q}}
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IC}{\mathbb{C}}
\DeclareMathOperator{\mer}{mer}
\DeclareMathOperator{\Sht}{Sht}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Et}{\acute{E}t}
\DeclareMathOperator{\et}{\acute{e}t}
\newcommand{\h}{\o{h}}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\etale}{\acute{e}tale}
\DeclareMathOperator{\Coker}{Coker}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\equi}{equi}
\DeclareMathOperator{\Hecke}{Hecke}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\HF}{HF}
\DeclareMathOperator{\HS}{HS}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\trdeg}{trdeg}
\DeclareMathOperator{\mod}{mod}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\log}{log}
\DeclareMathOperator{\Log}{Log}
\DeclareMathOperator{\Nm}{Nm}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Ob}{Ob}
\DeclareMathOperator{\Emb}{Emb}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\scale}{scale}
\DeclareMathOperator{\Sper}{Sper}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\Ét}{Ét}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\rad}{rad}
\DeclareMathOperator{\lim}{lim}
\DeclareMathOperator{\char}{char}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\length}{length}
\DeclareMathOperator{\locArt}{locArt}
\DeclareMathOperator{\***}{***}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\ker}{ker}
\DeclareMathOperator{\ht}{ht}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\Frac}{Frac}
\DeclareMathOperator{\det}{det}
\newcommand{\AA}{\sc{A}}
\newcommand{\Rem}{\gudl{\sc{R}\!emark}}
\newcommand{\Def}{\color{orange}{\underline{\color{black}{\sc{D}\!efinition}}}}
\newcommand{\Defn}[1]{\color{orange}{\underline{\color{black}{\sc{D}\!efinition\tx{}#1}}}}
\newcommand{\Prop}{\color{orange}{\underline{\color{black}{\sc{P}\!roposition}}}}
\newcommand{\Propn}[1]{\color{orange}{\underline{\color{black}{\sc{P}\!roposition\tx{}#1}}}}
\newcommand{\Claim}{\gudl{\sc{C}\!laim\colon}}
\newcommand{\Claimn}[1]{\gudl{\sc{C}\!laim \tx{}#1}}
\newcommand{\Thm}{\color{orange}{\underline{\color{black}{\sc{T}\!heorem}}}}
\newcommand{\Thmn}[1]{\gudl{\sc{T}\!heorem\tx{}#1}}
\newcommand{\O}{\c{O}}
\DeclareMathOperator{\Ouv}{Ouv}
\newcommand{\Cor}{\color{orange}{\underline{\color{black}{\sc{C}\!orollary}}}}
\newcommand{\Corn}[1]{\color{orange}{\underline{\color{black}{\sc{C}\!orollary\tx{}#1}}}}
\newcommand{\Fct}{\color{orange}{\underline{\color{black}{\sc{F}\!act}}}}
\newcommand{\Fctn}[1]{\color{orange}{\underline{\color{black}{\sc{F}\!act\tx{}#1}}}}
\newcommand{\Lem}{\color{orange}{\underline{\color{black}{\sc{L}\!emma}}}}
\newcommand{\Lemn}[1]{\color{orange}{\underline{\color{black}{\sc{L}\!emma\tx{}#1}}}}
\newcommand{\Exp}{\color{orange}{\underline{\color{black}{\sc{E}\!xample}}}}
\newcommand{\Expn}[1]{\color{orange}{\underline{\color{black}{\sc{E}\!xample\tx{}#1}}}}
\newcommand{\Rem}{\gudl{\sc{R}\!emark\colon}}
\newcommand{\Remn}[1]{\gudl{\sc{R}\!emark #1\colon}}
\newcommand{\brc}[1]{[\![#1]\!]}
\newcommand{\qst}{{}^{\color{red}{[?]}}}
\newcommand{\qstn}[1]{{}^{\color{red}{[?,#1]}}}
\newcommand{\sto}{\overset{\sim}{\to}}
\newcommand{\Ga}{\mathbb{G}_a}
\newcommand{\G}{\mathbb{G}}
\newcommand{\B}{\mathbb{B}}
\newcommand{\Gm}{\G_m}
\newcommand{\d}[1]{_{#1}}
\newcommand{\nz}{\not=0}
\newcommand{\x}{(x)}
\newcommand{\y}{(y)}
\newcommand{\r}[1]{\mid_{#1}}
\newcommand{\ij}{(i,j)}
\newcommand{\o}[1]{\operatorname{#1}}
\newcommand{\ne}{\not=\emptyset}
\newcommand{\ISLn}{\mathbb{S}\mathbb{L}_n}
\newcommand{\tfae}{\textbf{T.F.A.E.}}
\newcommand{\ndownlong}[2]{#1\ -\!\!\!\rightharpoonup\!\leftharpoondown\!\to\! #2}
\newcommand{\OC}{\c{O}_C}
\newcommand{\OF}{\c{O}_F}
\newcommand{\gsp}[1]{\udl{\Spec}_S(#1)}
\newcommand{\shso}{\udl{\text{Sheaves on}}}
\newcommand{\shs}{\udl{\text{Sheaves}}}
\newcommand{\ush}[1]{\udl{\text{Sheaf}}(#1)}
\newcommand{\sh}{\udl{\text{Sheaf}}}
\newcommand{\rr}{/\!\!/}
\newcommand{\EE}{\mathscr{E}}
\newcommand{\V}{\mathbb{V}}
\newcommand{\ddd}{(d,d_1,d_2)}
\newcommand{\Vd}{V_{d,d_1,d_2}}
\newcommand{\xy}{(x,y)}
\newcommand{\OX}{\c{O}_X}
\newcommand{\Ox}{\c{O}_{X,x}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\lims}{\limsup_{n\to \infty}}
\newcommand{\proof}{\gudl{\mathscr{P}\!roof}\colon}
\newcommand{\proofofprop}[1]{\underline{\color{orange}{\mathscr{P}\!roof\tx{}of\tx{}\sc{P}\!roposition\tx{}#1}\colon}}
\newcommand{\proofofcor}[1]{\underline{\color{orange}{\mathscr{P}\!roof\tx{}of\tx{}\sc{C}\!orollary\tx{}#1}\colon}}
\newcommand{\proofofthm}{\gudl{\sc{P}\!roof\tx{}of\tx{}\sc{T}\!heorem\colon}}
\newcommand{\proofofthmn}[1]{\gudl{\sc{P}\!roof\tx{}of\tx{}\sc{T}\!heorem\tx{}#1\colon}}
\newcommand{\Bew}{\underline{\color{orange}{\mathscr{B}\!eweis}\colon}}
\newcommand{\defeq}{\overset{\mathscr{D}\mathscr{e}\mathscr{f}.}{=\!=}}
\newcommand{\set}[2]{\{#1\mid #2\}}
\newcommand{\SS}{\mathscr{S}}
\newcommand{\FF}{\mathscr{F}}
\newcommand{\DD}{\mathscr{D}}
\newcommand{\dyadksum}[1]{\sum_{I\in \DD_k,I\sube J}#1}
\newcommand{\noem}{\not=\emptyset}
\newcommand{\DD}{\c{D}}
\newcommand{\BB}{\mathscr{B}}
\newcommand{\Pr}{\ff{P}}
\newcommand{\exact}[3]{0\to #1\to #2\to#3\to 0}
\newcommand{\qed}{\gudl{\ff{Q}.\ff{E}.\ff{D}.}}
\newcommand{\wt}[1]{\widetilde{#1}}
\newcommand{\wh}[1]{\widehat{#1}}
\newcommand{\spr}[1]{\Sper(#1)}
\newcommand{\LL}{\mathscr{L}}
\newcommand{\sp}[1]{\Spec(#1)}
\newcommand{\nuplong}[2]{#1\ -\!\!\!\rightharpoondown\!\leftharpoonup\!\to\! #2}
\newcommand{\ndownloong}[2]{#1 -\!\!\!-\!\!\!\rightharpoonup\!\leftharpoondown\!\!\!\longrightarrow \!#2}
\newcommand{\bop}{\bigoplus}
\newcommand{\eps}{\epsilon}
\newcommand{\K}{\mathbb{K}}
\newcommand{\lxen}{\langle x_1\cos x_n\rangle}
\newcommand{\Xen}{[X_1\cos X_n]}
\newcommand{\xen}{[x_1\cos x_n]}
\newcommand{\ip}{\langle -,- \rangle}
\newcommand{\ipr}[2]{\langle #1,#2 \rangle}
\newcommand{\vth}{\vartheta}
\newcommand{\pprod}{\prod_{v\in\ff{M}_\K}}
\newcommand{\pfam}[1]{(#1)_{v\in\ff{M}_\K}}
\newcommand{\finfam}[1]{(#1)_{i=1}^n}
\newcommand{\fam}[1]{(#1)_{i\in I}}
\newcommand{\jfam}[1]{(#1)_{j\in J}}
\newcommand{\kfam}[1]{(#1)_{k\in K}}
\newcommand{\nfam}[1]{(#1)_{i=1}^n}
\newcommand{\nifam}[1]{(#1)_{n=0}^\infty}
\newcommand{\udl}[1]{\underline{#1}}
\newcommand{\Uij}{U_i\cap U_j}
\newcommand{\vpi}{\varphi_i}
\newcommand{\vpj}{\varphi_j}
\newcommand{\vph}{\varphi}
\newcommand{\psij}{\psi_{i,j}}
\newcommand{\CC}{\c{C}}
\newcommand{\nsum}{\sum_{n\in\N}}
\newcommand{\twist}[1]{\c{O}_{\mathbb{P}_k^n}(#1)}
\newcommand{\prj}[1]{\Proj (#1)}
\newcommand{\part}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\kxn}{k[x_0,\pts,x_n]}
\newcommand{\ques}{\gudl{\c{Q}\!uestion\colon}}
\newcommand{\quesn}[1]{\gudl{\c{Q}\!uestion\tx{}#1\colon}}
\newcommand{\answ}{\gudl{\sc{A}\!nswer\colon}}
\newcommand{\cons}{\color{orange}{\udl{\color{black}{\sc{C}\!onsiderations:}}}}
\newcommand{\ka}{\kappa}
\newcommand{\pr}{\mathfrak{p}}
\newcommand{\abs}[1]{\left| #1\right|}
\newcommand{\ab}{\left|-\right|}
\newcommand{\eps}{\epsilon}
\newcommand{\N}{\mathbb{N}}
\newcommand{\KX}{K[X]}
\newcommand{\cov}{\c{U}}
\newcommand{\ff}[1]{\mathfrak{#1}}
\newcommand{\legendre}[2]{\left(\frac{#1}{#2}\right)}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\ANF}{K/\Q}
\newcommand{\GFF}{F/{\F_p(t)}}
\newcommand{\Os}{\mathcal{O}_{S,s}}
\newcommand{\lineb}{\sc{L}}
\newcommand{\cyclm}{\Q(\sqrt[m]{1})}
\newcommand{\cyclmK}{K(\sqrt[m]{1})}
\newcommand{\LX}{L[X]}
\newcommand{\GG}{\sc{G}}
\newcommand{\OS}{\mathcal{O}_S}
\newcommand{\bb}[1]{\textbf{#1}}
\newcommand{\OY}{\mathcal{O}_Y}
\newcommand{\vdp}{\sc{V}\!an\text{ }der\text{ }\sc{P}\!ut}
\newcommand{\weierstr***}{\sc{W}\!eierstraß}
\newcommand{\runge}{\sc{R}\!unge}
\newcommand{\laurent}{\sc{L}\!aurent}
\newcommand{\grothendieck}{\sc{G}\!rothendieck}
\newcommand{\noether}{\sc{N}\!oether}
\newcommand{\glX}{\Gamma(X,\mathcal{O}_X)}
\newcommand{\glY}{\Gamma(Y,\mathcal{O}_Y)}
\newcommand{\finKX}{f\in K[X]}
\newcommand{\ser}[1]{\sm{n=0}{\infty}{#1}}
\newcommand{\sm}[3]{\underset{#1}{\overset{#2}{\sum}} #3}
\newcommand{\cl}[1]{\overline{#1}}
\newcommand{\sube}{\subseteq}
\newcommand{\hk}{\hookrightarrow}
\newcommand{\OYy}{\mathcal{O}_{Y,y}}
\newcommand{\supe}{\supseteq}
\newcommand{\resy}{\kappa(y)}
\newcommand{\LK}{L/K}
\newcommand{\iso}{\overset{\sim}{\to}}
\newcommand{\isom}[3]{#1\overset{#2}{\iso}#3}
\newcommand{\kn}{k^n}
\newcommand{\kvec}{\textbf{vect}(k)}
\newcommand{\fkvec}{\textbf{vect}_{<\infty}(k)}
\newcommand{\fz}{f(X)=0}
\newcommand{\KIsom}{L\underset{K}{\overset{\sim}{\to}} L}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\L}{\mathbb{L}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\ad}{\A_k}
\newcommand{\P}{\mathbb{P}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Zp}{\mathbb{Z}_p}
\newcommand{\Qp}{\mathbb{Q}_p}
\newcommand{\Qq}{\mathbb{Q}_q}
\newcommand{\Fp}{\mathbb{F}_p}
\newcommand{\I}{[0,1]}
\newcommand{\In}{[0,1]^n}
\newcommand{\Fpn}{\mathbb{F}_{p^n}}
\newcommand{\Fpm}{\mathbb{F}_{p^m}}
\newcommand{\Zn}{\mathbb{Z}/{n\mathbb{Z}}}
\newcommand{\Zx}[1]{\mathbb{Z}/{#1\mathbb{Z}}}
\newcommand{\md}[3]{#1\equiv #2\pmod{#3}}
\newcommand{\ga}{\Gal(L/K)}
\newcommand{\aga}[1]{\Gal(\overline{#1}/#1)}
\newcommand{\sga}[1]{\Gal(#1^{sep}/{#1})}
\newcommand{\gal}[2]{\Gal(#1/{#2})}
\newcommand{\c}[1]{\mathcal{#1}}
\newcommand{\skw}{\{\tau\}}
\newcommand{\limes}[1]{\underset{i\in I}{\varprojlim{#1_i}}}
\newcommand{\IGLn}{\mathbb{G}\mathbb{L}_n}
\newcommand{\IGL}{\mathbb{G}\mathbb{L}}
\newcommand{\Co}[2]{H^p(#1,#2)}
\newcommand{\OK}{\mathcal{O}_K}
\newcommand{\OL}{\mathcal{O}_L}
\newcommand{\res}[1]{\kappa(#1)}
\newcommand{\resx}{\kappa(x)}
\newcommand{\lTen}{\langle T_1\cos T_n\rangle}
\newcommand{\lXen}{\langle X_1\cos X_n\rangle}
\newcommand{\Te}{[T]}
\newcommand{\Tee}{[T_1,T_2]}
\newcommand{\Teee}{[T_1,T_2,T_3]}
\newcommand{\Ten}{[T_1\cos T_n]}
\newcommand{\Tem}{[T_1\cos T_m]}
\newcommand{\pts}{\cdots}
\newcommand{\pt}{\cdot}
\newcommand{\hm}[3]{\Hom_{#1}(#2,#3)}
\newcommand{\hom}{\Hom}
\newcommand{\dash}{\dashrightarrow}
\newcommand{\schemes}{\bb{(Sch)}}
\newcommand{\groups}{\bb{(Grp)}}
\newcommand{\rings}{\bb{(Ring)}}
\newcommand{\tx}[1]{\text{ #1 }}
\newcommand{\mm}{\ff{m}}
\newcommand{\zkinfsum}{\sum_{k=0}^\infty}
\newcommand{\ziinfsum}{\sum_{i=0}^\infty}
\newcommand{\zjinfsum}{\sum_{j=0}^\infty}
\newcommand{\asum}[1]{\sum_{\a\in\N^n}#1 X^\a}
\newcommand{\arr}[3]{#1\overset{#2}{\to} #3}
\newcommand{\nrm}[1]{\left\|#1\right\|}
\newcommand{\nr}{\nrm{-}}
\newcommand{\ext}[2]{#1/{#2}}
\newcommand{\lam}{\lambda}
\newcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\g}{\gamma}
\newcommand{\de}{\delta}
\newcommand{\vp}{\varphi}
\newcommand{\p}{\phi}
\newcommand{\bul}{\bullet}
\newcommand{\t}{\tau}
\newcommand{\s}{\sigma}
\newcommand{\ze}{\zeta}
\newcommand{\T}{\mathbb{T}}
\newcommand{\tm}{\times}
\newcommand{\tms}{\times\pts\times}
\newcommand{\ot}{\otimes}
\newcommand{\ots}{\otimes\pts\otimes}
\newcommand{\pls}{+\pts +}
\newcommand{\cos}{,\pts,}
\newcommand{\op}{\oplus}
\newcommand{\ops}{\oplus\pts\oplus}
\newcommand{\cr}{\circ}
\newcommand{\crs}{\circ\pts\circ}
\newcommand{\sc}[1]{\mathscr{#1}}
\newcommand{\scal}[2]{\sc{#1}{\!#2}}
\newcommand{\ov}[2]{\begin{matrix}#1 \\ #2\end{matrix}}
\newcommand{\viele}{\color{orange}{\udl{\color{black}{\sc{V}\!iele\tx{}\sc{G}\!r\overset{{}_{,,\!}}{u}\textit{ß}e}}}}
\newcommand{\xst}{\color{orange}{\udl{\color{black}{X.S.T.\sim 小石头}}}}
\newcommand{\gudl}[1]{\color{orange}{\udl{\color{black}{#1}}}}
\newcommand{\Task}{\gudl{\sc{T}\!ask:}}
\newcommand{\Exer}{\gudl{\sc{E}\!exercise:}}
\newcommand{\Drinfeld}{\gudl{\sc{D}\!rinfeld:}}
\newcommand{\Goss}{\gudl{\sc{G}\!oss}}
\newcommand{\CK}{C/K}
\newcommand{\CS}{C/S}
\newcommand{\Ck}{C/k}
\newcommand{\Om}{\Omega}
\newcommand{\J}{\Jac_{\CS}^{g-1}}
\newcommand{\Fact}{\gudl{\sc{F}\!act\colon}}
\newcommand{\Factn}[1]{\gudl{\sc{F}\!act\tx{}#1\colon}}\)
Hallo!
Es geht nochmal darum eine Galois-Gruppe zu bestimmen, und zwar folgenden Polynoms:
$X^8 - 2 \in \IQ[X]$.
---
Der Zerfällungskörper ist $\IQ(\sqrt[8]{2}, i)$ und die Erweiterung hat den Grad 16.
Es gilt:
$\text{Gal}(\IQ(\sqrt[8]{2}, i) / \IQ) \subset S_8$.
---
Ich komme hier aber nicht weiter, wie kann ich nun die 16 Automorphismen der Galois-Gruppe bestimmen?
Danke!
----------------- Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄\(\endgroup\)
|
Notiz Profil
Quote
Link |
ollie3
Aktiv  Dabei seit: 21.02.2016 Mitteilungen: 62
 |     Beitrag No.1, eingetragen 2020-10-23
|
Hallo,
ich glaube nicht, das das so richtig
ist. Müsste nicht die prim.8. Einheitswuzel
mit im zerfällunskörper sein?
|
Notiz Profil
Quote
Link |
ollie3
Aktiv  Dabei seit: 21.02.2016 Mitteilungen: 62
 |     Beitrag No.2, eingetragen 2020-10-23
|
Sorry,
das war ein Denkfehler, der zerfällungskörper
war doch richtig...
|
Notiz Profil
Quote
Link |
Triceratops
Aktiv  Dabei seit: 28.04.2016 Mitteilungen: 5461
Herkunft: Berlin
 |     Beitrag No.3, eingetragen 2020-10-23
|
Notiz Profil
Quote
Link |
Triceratops
Aktiv  Dabei seit: 28.04.2016 Mitteilungen: 5461
Herkunft: Berlin
 |     Beitrag No.4, eingetragen 2020-10-23
|
@ollie3: Das stimmt schon, aber die primitive 8. Einheitswurzel ist $$\sqrt{i} = \frac{1+i}{\sqrt{2}} = \frac{1+i}{\sqrt[8]{2}^4},$$sodass man genauso gut $i$ als Erzeuger nehmen kann.
|
Notiz Profil
Quote
Link |
LukasNiessen
Aktiv  Dabei seit: 30.09.2019 Mitteilungen: 148
Herkunft: Nordrhein-Westfalen, Bonn, Weststadt
 |     Beitrag No.5, vom Themenstarter, eingetragen 2020-10-26
|
Perfekt, danke euch!
----------------- Beste Grüße, Lukas Nießen
PS: Schreibt mir gerne 😄
|
Notiz Profil
Quote
Link |
LukasNiessen hat die Antworten auf ihre/seine Frage gesehen. LukasNiessen hat selbst das Ok-Häkchen gesetzt. | [Neues Thema] [Druckversion] |
|
All logos and trademarks in this site are property of their respective owner. The comments are property of their posters, all the rest © 2001-2021 by Matroids Matheplanet
This web site was originally made with PHP-Nuke, a former web portal system written in PHP that seems no longer to be maintained nor supported. PHP-Nuke is Free Software released under the GNU/GPL license.
Ich distanziere mich von rechtswidrigen oder anstößigen Inhalten, die sich trotz aufmerksamer Prüfung hinter hier verwendeten Links verbergen mögen. Lesen Sie die
Nutzungsbedingungen,
die Distanzierung,
die Datenschutzerklärung und das Impressum.
[Seitenanfang]
|