Sharedwww / qual_syllabus.dviOpen in CoCalc
����;� TeX output 1997.08.19:1509������y�����?������5��D��tG�G�cmr17�Qualifying�7tExam�Syllabus�����������X�Qcmr12�William��Stein������%��M��"V

cmbx10�Committee:�٠�K�`y

cmr10�R.��AHartshorne�(Chair),��<H.W.�Lenstra,�V.�Sergano���v��q�a,�Bin�Y��*�u����>(Statistics)��!�>���N�ffcmbx12�1��VL�Algebraic�ffNum���b�s3er�Theory�����>�References:�\bCassels-F��*�rohlic���h�*���':

cmti10�A���lgebr��}'aic�l�Numb�er�The�ory�,�3Ch�*�I-I�GI;�Lang��A���lgebr��}'aic����>Numb��}'er���The�ory�,�UUCh�VI�GI�I.�������M�
!",�

cmsy10�����W�Algebraic��TNum��9b�Q�er�Fields.�������[email protected]{����m�Obje��}'cts��jattache�d�to�a�numb�er�eld��
�b>

cmmi10�K���:����discriminan���t��h�disc��!�(�K��),��5dieren�t,����mring��Aof�in���tegers,���units,�real��Aand�complex�places,���ideal�class�group,����madeles��#�A����	0e�rcmmi7�K�����,��Videles��J����K���,��Varc���himedian�and�nonarc�himedian�v��q�aluations,����mRiemann's�UUzeta�function.�������[email protected]�{����m�Standar��}'d�(�facts:�7��the��ring�of�in���tegers�is�a�Dedekind�domain,��IDiric�hlet's����munit�UUtheorem,�the�class�group�is�nite.�������[email protected]�{����m�Applic��}'ations���of�standar�d�to�ols�(not�pr�o�ofs):���Analytic�]wclass�n���um�b�Ger����mform���ula,�2�Cheb�Getaro�v's�*Densit�y�Theorem,�2�Kronec�k�er-W��*�eb�Ger�theorem,����mClass�UUField�Theory��*�.�������M�����W�Lo�Q�cal��TFields.�������[email protected]{����m�Structur��}'e�:��#unramied,�1�tamely��ramied,�and�totally�ramied�exten-����msions,�UUF��*�rob�Genius�automorphisms.�������[email protected]�{����m�Standar��}'d�H�facts�:�H��ef�ڧ�=���n�,�Hensel's��lemma,�nite�unramied�extensions����mof�2k�Q����p��ѽ�corresp�Gond�bijectiv���ely�to�nite�extensions�of��F����p���R�,�i�the�Brauer����mgroup�UUof�a�lo�Gcal�eld.�������M�����W�Examples.��Q�,���Q�(�������p���UW�����fe���wv��D������),��Q�(�����p���R�),�cubic���elds,�biquadratic�elds.��Explicit����Wcomputation�W�of�discriminan���t,��1dieren�t,�class�W�group,�units,�in�tegers,�and����Wgalois�UUgroup�of�a�n���um�b�Ger�UUeld�of�lo���w�degree.�������M�����W�Applications.�qDzBasic�UUco�Gding�theory�and�cryptograph���y��*�.������1����*�y�����?�������>�2��VL�Algebraic�ffGeometry�����>�Reference:�q�Hartshorne,�UU�A���lgebr��}'aic���Ge�ometry�,�Ch�I-I�GI.�������M�����W�V��
�arieties�Vo��9v�er�an�algebraically�closed�eld.�L�ane��and�pro��8jectiv���e�v��q�a-����Wrieties,��Dco�Gordinate�j?rings,�function�elds,�rational�maps,�nonsingular�curv���es,����WNullstellensatz.�������M�����W�Sc��9hemes.�������[email protected]{����m�Basics:��E�shea���v�es,�Jsp�Gectrum�of�a�ring,�ane�sc���hemes,�sc�hemes,�mor-����mphisms�UUb�Get���w�een�sc�hemes.�������[email protected]�{����m�Pr��}'op�erties��8of�Schemes:�㎲regular,�q�normal,�in���tegral,�reduced,�irreducible,����mno�Getherian.�������[email protected]�{����m�Pr��}'op�erties��Bof�Morphisms:��separated,��4nite���t���yp�Ge,�nite,�ane,�prop�er,����m
at.�������[email protected]�{����m�She��}'aves:�0زcoheren���t��vand�quasi-coheren�t�shea�v�es,��pin�v�ertible�shea�v�es,��pthe����mfunctors�UU�f�����^��O!�cmsy7���	Ȳ(pullbac���k),��f�������9�(pushforw�ard).�������[email protected]�{����m�Divisors:�qDzPicard�UUgroup,�Cartier�divisors,�W��*�eil�divisors.��!č��>�3��VL�Lie�ffAlgebras�����>�References:�q�Humphreys,�UU�Lie���A���lgebr��}'as�and�R�epr�esentation�The�ory�,�UUc���h�I-I�GI�I��������[email protected]�{����m�Basic��adenitions:��W�Lie��algebra,���the�classical�Lie�algebras,�deriv���ed�se-����mries,�UUlo���w�er�cen�tral�series,�radical.�������[email protected]�{����m�Pr��}'op�erties���of�Lie�algebr��}'as:�qDzsolv��q�able,�UUnilp�Goten���t,�semisimple,�simple.�������[email protected]�{����m�The��}'or�ems:�qDzLie's�UUtheorem,�Engel's�theorem,�Cartan's�criterion.�������[email protected]�{����m�Corr��}'esp�ondenc�e:��p�the�V�corresp�Gondence�b�et���w�een�V�simple�Lie�algebras�and����mirreducible�UUro�Got�systems,�Killing�form,�Cartan�decomp�osition.�������[email protected]�{����m�Classic��}'ation���of�simple�r�o�ot�systems:�qǵA����n��q~�,�UU�B����n���,��C����n���,�etc.������2����	q���;�y���':

cmti10���N�ffcmbx12��"V

cmbx10�X�Qcmr12�D��tG�G�cmr17�
!",�

cmsy10�O!�cmsy7�
�b>

cmmi10�	0e�rcmmi7�K�`y

cmr10�K�������