Galois Theory

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You will learn to compute Galois groups and before that study the properties of various field extensions. We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra finite algebras over a field, base change via tensor product and apply this to study the notion of separability in some detail.

After that we shall discuss Galois extensions and Galois correspondence and give many examples cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc. We shall address the question of solvability of equations by radicals Abel theorem.

We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings integral elemets, norms, traces, etc. Some knowledge of commutative algebra prime and maximal ideals — first few pages of any book in commutative algebra is welcome.

Galois Theory

For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally, the statement of Sylow's theorems. There will be two non-graded exercise lists in replacement of the non-existent exercise classes Do you have technical problems? Write to us: coursera hse. Introduction -This is just a two-minutes advertisement and a short reference list. Week 1 -We introduce the basic notions such as a field extension, algebraic element, minimal polynomial, finite extension, and study their very basic properties such as the multiplicativity of degree in towers.

Week 2 -We introduce the notion of a stem field and a splitting field of a polynomial. Using Zorn's lemma, we construct the algebraic closure of a field and deduce its unicity up to an isomorphism from the theorem on extension of homomorphisms. Week 3 -We recall the construction and basic properties of finite fields.

We prove that the multiplicative group of a finite field is cyclic, and that the automorphism group of a finite field is cyclic generated by the Frobenius map. We introduce the notions of separable resp. We briefly discuss perfect fields. This week, the first ungraded assignment in order to practice the subject a little bit is given.

An Introduction to Galois Theory

Week 4 -This is a digression on commutative algebra. The notion of a solvable group in group theory allows one to determine whether a polynomial is solvable in radicals, depending on whether its Galois group has the property of solvability. If all the factor groups in its composition series are cyclic, the Galois group is called solvable , and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field usually Q.

By the rational root theorem this has no rational zeroes. Neither does it have linear factors modulo 2 or 3. Thus its modulo 3 Galois group contains an element of order 5. It is known [11] that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals.

A permutation group on 5 objects with elements of orders 6 and 5 must be the symmetric group S 5 , which is therefore the Galois group of f x.

Geometry of Field Extensions

This is one of the simplest examples of a non-solvable quintic polynomial. According to Serge Lang , Emil Artin found this example. As long as one does not also specify the ground field , the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field K and a finite group G. Cayley's theorem says that G is up to isomorphism a subgroup of the symmetric group S on the elements of G. G acts on F by restriction of action of S. On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers.

Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q. Various people have solved the inverse Galois problem for selected non-Abelian simple groups. Existence of solutions has been shown for all but possibly one Mathieu group M 23 of the 26 sporadic simple groups.

There is even a polynomial with integral coefficients whose Galois group is the Monster group. From Wikipedia, the free encyclopedia. Main article: Inverse Galois problem. Galois Theory. Chapman and Hall. Galois' Theory of Algebraic Equations. World Scientific.

Elements of Abstract Algebra. Courier Corporation. Braunschweig, Vieweg. European Mathematical Society. Prasolov, Polynomials , Theorem 5. Students on this programme can substitute a one 10 credit MATH optional course unit for a University College course unit.

1.2 History

Students who choose this option will need to complete the course unit permission form available under the section Forms. In addition, student must choose PHIL optional course units worth 40 credits listed below , totalling credits. Please check the timetables for these course units before selection. Students who choose this option must complete the online course unit permission form available from the section Forms.

Please check the timetables before selecting course units and the level of the course unit.

Plus 30 credits of MATH optional course units in semester 2, totalling credits. To take an outside course unit you will need the permission from the Year Tutor and the course unit permission form to be completed. The third year of this programme consists of credits of optional course units. At least credits must be taken from level 3 or level 4 course unit options from the list available from the section on course units including at least 80 credits of level 3 or level 4 MATH options.

Up to 40 credits can be taken from the approved non-MATH course units and a maximum of 20 credits of level 2 course units may be taken. In the case of language course units, the first digit of the course unit code refers to the language level, not to the academic level, so students may take language course units whose first digit is 1 or 2 and these course units do not count towards the 20 credits of level 2 course unit allowed in the third year.

Students should consider taking up to two Level 4 MATH course unit options in the third year as this will increase choice and give flexibility in the fourth year. Plus 90 credits of Level 4 MATH course units list of level 4 course units available from the section on course units or if you took level 4 MATH course units in the third year, the necessary amount to ensure that you have done at least credits of level 4 MATH course units in years 3 and 4 combined.

Up to 20 credits can be taken from the approved level 3 non MATH course units with permission if you took level 4 course units in year 3. The Department of Mathematics offers projects to students in their third and fourth years. The projects can be a two-semester double project, or single semester projects.

The MATH double project for third year students is weighted 20 credits and MATH double project for fourth year students is weighted 30 credits. Students who wish to take the latter option are required to arrange supervisors for both projects at the start of the academic year. There is a possibility of one combined project in an appropriate topic in mathematical physics with permission from both the Mathematics and Physics project coordinators.

Students may opt to do a project, which may be either a one semester 10 credit or a two semester 20 credit project. A first semester project may be converted to a two semester project up to the ninth week of the first semester, and only with the agreement of the supervisor.

Free Online Course: Introduction to Galois Theory from Coursera | Class Central

On some joint programmes where the mathematics modules of the third year would amount to less than 40 credits, you may not be permitted to take a double mathematics project. Course requirements Browse the units below for the current academic year, and find prerequisites and material links. Selecting your course units Students continuing to their second, third or fourth year of studies will need to select course unit options on the student system. How to select your course units Please note: Course unit selection will then re-open for the first two weeks of semester 1, and the same for semester 2.

Course unit selection checklist Step 1: Check the programme requirements Students are required to take a number of mandatory and optional course units as part of their programme of study. Step 2: Do your research The course units are split into two categories: mandatory course units and optional course units. Some course units are capped due to the nature of the course assessment and fill quite quickly. Step 4: Check if approval is required Some course units have restrictions on numbers or a separate approval process.

Lecture 7: Introduction to Galois Fields for the AES by Christof Paar

Students interested in enrolling onto this course unit will need to complete an on-line registration process whereby students will need to write no more than words as to why they wish to take this course unit. The registration form is available from the forms section electronically. The application to register will open and close the same time as the course unit selection process. Once your application has been approved you will be informed by email before the academic year starts. During the first lecture you will be asked to apply for a DBS Disclosure Barring Services check from the post office which you can be reimbursed for by completing a PR7 claim form available from reception in the Alan Turing building.

Indicative project titles and supervisors will be available in the study module early July.

Submission history

Note some projects will have been allocated to at least one MMath student and may have limited availability. If you have queries about specific projects please contact the supervisor directly. The application to register for a project will open during course unit selection but will remain open until the end of week 1 of semester 1. Further details are available from the programme information link below. There is a registration process for enrolling on these course units which is explained in detail in the Undergraduate Project Guide from the project information link below.

Outside course units - The Single Honours programmes allow students to choose a subject other than mathematics. Students will need to check their programme requirements as to whether they can choose outside course units as part of their programme of study.