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MTH 610(sec 201) Galois Theory Syllabus Spring 2012 CRN 4307 Prerequisites: MTH 450 and MTH 452 , also must have taken MTH 300 Course Objectives : To learn about Galois groups on extension fields, splitting fields and solvability of polynomial equations. Meeting time : T , R 12:30-1:45pm Room 518 Smith Hall Instructor : Dr. Alan Horwitz Office : Room 741 Smith Hall Phone : (304)696-3046 Email : [email protected] Text : Galois Theory, 3rd edition, Ian Stewart, Chapman and Hall/CRC Press Grading : homework (and possibly labs) 35% (175points) 2 major exams 40% (200 points) (if we have a 3rd exam, then I'll count the highest two) final( comprehensive ) exam 25% (125 points) total : 500 points Final exam date : Thursday May 1, 2012 at 12:45-2:45 pm General Policies : Attendance is required . You are responsible for reading the text, working the exercises, coming to office hours for help when you’re stuck, and being aware of the dates for homework assignments and major exams as they are announced. Late homework will be penalized. Exam dates will be announced at least one week in advance. Makeup exams will be given only if you have an acceptable written excuse with evidence and/or you have obtained my prior permission. Make ups are likely to be more difficult than the original exam and must be taken within one calendar week of the original exam date. You can’t make up a makeup exam: if you miss your appointment for a makeup exam, then you’ll get a score of 0 on the exam. If you anticipate being absent from an exam due to a prior commitment, let me know in advance so we can schedule a makeup. If you cannot take an exam due to sudden circumstances, you must call my office and leave a message on or before the day of the exam! In borderline cases, your final grade can be influenced by factors such as your record of attendance, whether or not your exam scores have been improving during the semester, and your class participation. Attendance: Regular attendance is expected ! If your grade is borderline, then good attendance can result in attaining a higher grade. Likewise, poor attendance can result in a lower grade. Having more than 3 weeks worth of unexcused absences (i.e., 6 lectures ) will automatically result in a course grade of F! Being habitually late to class will count as an unexcused absence for each occurrence. Carrying on conversations with your neighbor could be counted as being absent. Walking out in the middle of lecture is rude and a distraction to the class ; each occurrence will count as an unexcused absence. If you must leave class early for a doctor’s appointment , etc., let me know at the beginning and I’ll usually be happy to give permission. Absences which can be excused include illness, emergencies, or official participation in another university activity. Documentation from an outside source ( eg. coach, doctor, court clerk…) must be provided. If you lack documentation, then I can choose whether or not to excuse your absence. Sleeping in Class : Habitual sleeping during lectures can be considered as an unexcused absence for each occurrence. If you are that tired, go home and take a real nap! You might want to change your sleeping schedule, so that you can be awake for class. Cell Phone and Pager Policy : Unless you are a secret service agent, fireman, paramedic on call or a drug dealer, all electronic communication devices such as pagers and cell phones should be shut off during class. Alternatively, you can use the heel of your boot to grind your cell phone into the linoleum. Violation of this policy can result in confiscation of your device and your forced participation in a study on the deleterious health effects of excessive cell phone use. Week 1 Dates Spring 2012 1/91/13 2 1/171/20 3 1/231/27 4 1/302/3 5 2/62/10 Sections covered and topics Week 7 Dates Spring 2012 2/202/24 8 2/273/2 9 3/53/9 10 3/123/16 (last day to drop on 3/16) 11 3/263/30 Sections covered and topics 12 4/2-4/6 (Assessment day on 4/6) Week Dates Spring 2012 13 4/94/13 14 4/16-4/20 15 4/234/27 Sections covered and topics MTH 610(sec 201) Topics in Galois Theory(Ian Stewart) 1/10/12 1.1 complex numbers 1.2 review of groups, rings, fields, subrings and subfields, homomorphisms and isomorphisms monomorphisms vs. automorphisms nth-roots of unity 1.3 solving equations as a motivation for new number systems 1.4 solving quadratic equations by completing the square using Tschirnhaus transformations to rewrite quadratic, cubic, quartic, quintic equations Cardano’s Formula for solving cubic equations and the strange form of it solutions solving quartic equations why Tschirnhaus transformation method does not work for solving quintic equations 2.1 properties of polynomials over complex numbers polynomials in several variables notation for degree of a polynomial in one variable 2.2 Fundamental Theorem of Algebra winding number of a loop in the plane stays constant under slight deformations proving the Fundamental Theorem of Algebra 2.3 Remainder Theorem completely factoring any complex polynomial over the complex numbers 3.1 Division Algorithm greatest common factor of two polynomials using the Euclidean Algorithm to find a greatest common factor expressing greatest common factor as a linear combination of the two polynomials 3.2 reducible vs. irreducible polynomials over a subring polynomials can be factored as a product of irreducible polynomials coprime polynomials have greatest common factor of 1 uniqueness of factorizations into product of irreducible polynomials 3.3 Gauss Lemma on irreducibility over integers and rational numbers 3.4 Eisenstein’s Criterion for irreducibility over the rationals 3.5 Congruence modulo n Units in integers modulo n Euler phi function Using irreducibility in modulo n to show irreducibility over the integers 3.6 zeros of multiplicity m for polynomials the number of distinct zeros is no larger than the degree 4.1 field extension monomorphism subfield generated by a subset of a field subfield generated by adjoining a subset of a field extension 4.2 4.3 5.1 5.2 5.3 rational expressions in one and in n variables simple field extensions commutative diagram to represent an isomorphism between two field extensions algebraic over a field vs. transcendental over a field monic polynomials minimal polynomial of an algebraic element in a field extension any irreducible monic polynomial is a minimal polynomial for some complex number reduced form of a polynomial modulo m ideals generated by irreducible polynomials give quotient rings which are fields MTH 610(sec 201) Topics in Galois Theory(Ian Stewart) 1/10/12_____________________________________________________________________________ 5.4 simple algebraic extensions are isomorphic to quotient rings generated by a minimal polynomial algebraic extensions with same minimal polynomial are isomorphic degree of a simple algebraic extension matches degree of the minimal polynomial extending an isomorphism between fields to an isomorphism between simple algebraic extensions of the fields 6.1 viewing a field extension as a vector space over the field the degree of a field extension over a field 6.2 the short and regular Tower Laws for degrees of field extensions over subfields degree of a simple transcendental extension over its field is infinite degree of a simple algebraic extension over a field is equal to the degree of the minimal polynomial finite extensions are algebraic extensions formed by attaching finitely many algebraic elements 7.1 an angle can be trisected with a compass and a marked ruler definition of a point being constructible using a compass and unmarked ruler from an original point tower of subfields created by adjoining coordinates of constructible points coordinates of constructible points are zeros of quadratic polynomials fields created by adjoining constructible points have degrees which are powers of 2 over the field generated by the coordinates of the original point 7.2 Wantzel’s Theorem on not being able to “duplicate” a cube with ruler and compass constructions Wantzel’s Theorem on not being able to trisect a 60 degree angle with ruler and compass constructions being unable to square a circle with ruler and compass constructions 8.2 permutations of the zeros preserve the coefficients of the polynomial equation and form a subgroup of a symmetric group the Galois group of zeros of a polynomial equation 8.3 describing subgroups of a Galois group by the fields which they leave fixed subgroups of Galois group determine whether the polynomial equation can be solved by radicals 8.4 K-automorphisms of a field extension L Galois correspondence between subgroups of a Galois group and subfields of the field extension containing all zeros(which correspond to the Galois group) 8.5 all K-automorphisms of a field extension are defined as the Galois group 8.6 fixed fields of subgroups of the Galois group Galois correspondence between subgroups of a Galois group and their fixed fields 8.7 writing coefficients of a polynomial as elementary symmetric polynomials of the zeros tower of subfields needed for solubility by Ruffini radicals refining a tower of subfields to make degrees of consecutive field extensions prime lemmas on cyclic quotient groups of symmetric group and alternating group general polynomials of degree 5 or higher are insoluble by Ruffini radicals Klein 4-group the discriminant of a polynomial 8.8 solvability by radicals implies solubility by Ruffini radicals general polynomial equations of degree 5 or higher are not solvable by radicals theorem of natural irrationalities the height of a radical extension over a field is always finite factorability of the minimal polynomial of an element in an extension field MTH 610(sec 201) Topics in Galois Theory(Ian Stewart) 1/10/12_____________________________________________________________________________ 9.1 splitting field of a polynomial isomorphisms between fields extend to isomorphisms between splitting fields 9.2 ……………..to be continued………………..