## Attachment 'NF_SEP.tex'

Downloadlatex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=latex) entering extended mode (./latex_c3b7541409ae8736db48e05629866521ac5b4dbf_p.tex LaTeX2e <2014/05/01> Babel <3.9l> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def (/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu) (/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu) (/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu))) No file latex_c3b7541409ae8736db48e05629866521ac5b4dbf_p.aux. ! LaTeX Error: Can be used only in preamble. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.7 \documentclass [12pt]{article} ! LaTeX Error: Can be used only in preamble. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.9 \usepackage {amssymb, amsmath} ! LaTeX Error: Can be used only in preamble. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.17 \begin{document} [1] ! Undefined control sequence. <recently read> \mathbb l.30 ...by an irreducible polynomial over $\mathbb {Q}$, a class for number f... ! Undefined control sequence. \Gal ->\operatorname {Gal} l.30 ...er field $L/K$ and a subgroup $H$ of $\Gal (L/K)$, as well as others.... LaTeX Warning: Reference `NF' on page 2 undefined on input line 30. [2] ! Undefined control sequence. <recently read> \mathbb l.32 ...ce they are simultaneously rings, $\mathbb {Z}$-modules, and in the c... ! Undefined control sequence. <recently read> \mathbb l.32 ...s: does one represent elements as $\mathbb {Z}$-linear combinations o... ! Undefined control sequence. <recently read> \mathbb l.32 ...thbb{Z}$-linear combinations of a $\mathbb {Z}$ basis for the order, ... LaTeX Warning: Reference `Orders' on page 3 undefined on input line 32. ! Undefined control sequence. <recently read> \mathfrak l.34 .... For an ideal $I$ of an order $\mathfrak {O}$ over a maximal order ... ! Undefined control sequence. <recently read> \mathfrak l.34 ...n give generators for $I$ as an $\mathfrak {O}$ module, generators as... ! Undefined control sequence. <recently read> \mathbb l.34 ...hcal{O}_K$ module or a basis as a $\mathbb {Z}$ module. We also want... LaTeX Warning: Reference `Ideals' on page 3 undefined on input line 34. LaTeX Warning: Reference `UC' on page 3 undefined on input line 36. ! Undefined control sequence. \Gal ->\operatorname {Gal} l.38 ...e able to work with the Galois group $\Gal (L/K)$ not just as an abst... LaTeX Warning: Reference `Gal' on page 3 undefined on input line 38. ! Undefined control sequence. <recently read> \mathfrak l.40 A number field $L/K$ and a place $\mathfrak {w}$ of $L$ over $\mathfrak{... ! Undefined control sequence. <recently read> \mathfrak l.40 ...lace $\mathfrak{w}$ of $L$ over $\mathfrak {v}$ of $K$ determine a co... ! Undefined control sequence. l.40 ...f $K$ determine a completion $L_{\mathfrak {w}}$ over $K_{\mathfrak{v... ! Undefined control sequence. l.40 ...tion $L_{\mathfrak{w}}$ over $K_{\mathfrak {v}}$. Sage should suppor... LaTeX Warning: Reference `Comp' on page 3 undefined on input line 40. [3] ! Undefined control sequence. <recently read> \mathbb l.50 ...d than Sage, they do not consider $\mathbb {Q}$ a number field. But ... Overfull \hbox (73.4217pt too wide) in paragraph at lines 52--53 []\OT1/cmr/m/n/12 On the par-ent level, \OT1/cmtt/m/n/12 NumberField[]generic \ OT1/cmr/m/n/12 will have sub-classes \OT1/cmtt/m/n/12 NumberField[]absolute\OT1 /cmr/m/n/12 , Overfull \hbox (132.28838pt too wide) in paragraph at lines 52--53 \OT1/cmtt/m/n/12 NumberField[]relative\OT1/cmr/m/n/12 , \OT1/cmtt/m/n/12 Number Field[]multiple\OT1/cmr/m/n/12 , \OT1/cmtt/m/n/12 NumberField[]subfield\OT1/cmr /m/n/12 , \OT1/cmtt/m/n/12 NumberField[]fixed[]field ! Undefined control sequence. <recently read> \mathbb l.54 ...ingle irreducible polynomial over $\mathbb {Q}$. I'm not totally con... ! Undefined control sequence. <recently read> \mathbb l.54 ...ve extensions that happen to have $\mathbb {Q}$ as the ground field (... ! Undefined control sequence. <recently read> \mathbb l.54 ...eral (no unramified extensions of $\mathbb {Q}$, etc. Thoughts here ... [4] Overfull \hbox (35.1587pt too wide) in paragraph at lines 60--61 \OT1/cmtt/m/n/12 NumberField[]relative\OT1/cmr/m/n/12 ; for $\OML/cmm/m/it/12 m > \OT1/cmr/m/n/12 1$ it will be very sim-i-lar to \OT1/cmtt/m/n/12 NumberField []multiple\OT1/cmr/m/n/12 , Overfull \hbox (2.58733pt too wide) in paragraph at lines 60--61 \OT1/cmr/m/n/12 quired for \OT1/cmtt/m/n/12 NumberField[]multiple\OT1/cmr/m/n/1 2 ... How-ever, \OT1/cmtt/m/n/12 NumberField[]subfield \OT1/cmr/m/n/12 comes ! Undefined control sequence. \Gal ->\operatorname {Gal} l.62 ...l extensions?) and a subgroup $H$ of $\Gal (L/K)$, create the fixed s... Overfull \hbox (64.13467pt too wide) in paragraph at lines 66--67 \OT1/cmr/m/n/12 though both types should pre-sum-ably in-herit from a sin-gle \ OT1/cmtt/m/n/12 NumberFieldElement[]generic [5] ! Undefined control sequence. <recently read> \mathbb l.72 ...dule. In particular, orders over $\mathbb {Z}$ will be free $\mathbb... ! Undefined control sequence. <recently read> \mathbb l.72 ...rs over $\mathbb{Z}$ will be free $\mathbb {Z}$ modules. Overfull \hbox (131.86336pt too wide) in paragraph at lines 74--75 \OT1/cmr/m/n/12 of or-ders. It's not clear to me how it would be use-ful to hav e an \OT1/cmtt/m/n/12 NumberFieldOrder[]abelian[]extension Overfull \hbox (5.08522pt too wide) in paragraph at lines 74--75 \OT1/cmr/m/n/12 class, but all of the other cor-re-spond-ing classes, \OT1/cmtt /m/n/12 NumberFieldOrder[]absolute\OT1/cmr/m/n/12 , Overfull \hbox (80.21031pt too wide) in paragraph at lines 74--75 \OT1/cmtt/m/n/12 NumberFieldOrder[]relative\OT1/cmr/m/n/12 , \OT1/cmtt/m/n/12 N umberFieldOrder[]multiple\OT1/cmr/m/n/12 , \OT1/cmtt/m/n/12 NumberFieldOrder[]s ubfield [6] [7] (./latex_c3b7541409ae8736db48e05629866521ac5b4dbf_p.aux) LaTeX Warning: There were undefined references. LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. ) (\end occurred inside a group at level 1) ### semi simple group (level 1) entered at line 17 (\begingroup) ### bottom level (see the transcript file for additional information) Output written on latex_c3b7541409ae8736db48e05629866521ac5b4dbf_p.dvi (7 pages , 20564 bytes). Transcript written on latex_c3b7541409ae8736db48e05629866521ac5b4dbf_p.log.

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