https://studiegids.vu.nl/en/courses/2024-2025/XB_0134The student knows the following concepts, and can solve problems about and with them in explicit situations:basic field theory (various properties of field extensions, including finiteness, algebraicity, separability and normality);splitting fields, finite Galois extensions and the Galois correspondence;cyclotomic extensions of the rationals;finite fields, as well as their finite extensions together with their Galois groups.The student knows the criteria for constructibility by straightedge and compass, and solvability by radicals, and can apply those in explicit situations.The Babylonians could solve quadratic equations, but it took until the 16th century before (complicated) formulae were discovered for solving cubic and quartic equations. These use iterated expressions of (varying) n-th roots ('radicals'). The quest for similar formulae for higher degree equations culminated in a negative answer by Ruffini and Abel in the early 19th century. Galois soon after refined the result, providing a precise criterion in terms of the symmetries of the zeroes of the equation that determines if those zeroes can be expressed using radicals. After a more general study of field extensions and their properties, we discuss the fundamental theorem of Galois theory, one of the most beautiful results in algebra. We also discuss its consequences for the (in)solvability of the quintic by radicals, and for the (in)constructibility of regular n-gons by means of straightedge and compass. We treat the following topics.Properties of field extensions (including finite, algebraic, simple, separable, inseparable, normal or finite Galois extensions); examples.Compositum of fields; algebraic closure (including a different proof of the fundamental theorem of algebra, which asserts that the complex numbers are algebraically closed).Cyclotomic fields.The fundamental theorem of Galois theory, which gives a 1-1 correspondence between the subgroups of the Galois group of a finite Galois extension, and the intermediate fields.Finite fields, as well as their finite extensions together with their Galois groups.Constructibility using straightedge and compass; (in)solvability of polynomials using radicals.Lectures (14x2 hours) and tutorials (14x2 hours)Two partial exams (or a resit), and marked assignments. The assignments in total count for 5% towards the grade, the average of scores for the two partial exams (or the score of the resit) counts for 95%. Alternatively, the result of the exam or the resit counts for 100% of the grade if this results in a higher score. All of this is subject to the requirement that the average of the scores for the two partial exams (or the score of the resit) is at least 55%, otherwise the grade is capped at 5.David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition (2003), John Wiley and Sons.Bachelor Mathematics Year 2The VU courses Linear Algebra, Group Theory, as well as Rings and fields. The results treated in these three courses are used extensively throughout this course.