Learning Outcomes
Upon successful completion of the course students will
i. will determine whether an extesnsion is simple
ii. will determine the degree and the minimal polynomial of an extension
iii. will construct the Galois group of an extension and determine its subgroups
iv. will determine the intetermediate fields of an extension
v. will recognize the correspondence between the subgroups of a Galois group and the intermediate fields of a normal extension
vi. will apply the results of Galoi theory for the solvability of polynomials
vii. will apply the results of Galoi theory on constructions by ruler and compass
Course Content (Syllabus)
Field extensions. Prime fields. Algebraic and Transcendental extensions. Classification of simple extensions. Constructions with ruler and compass. Algebraic closure of a field. Splitting fields. Normal and Separable extensions. Finite fields. Automorphisms of fields. Galois group and Galois extension. Fundamental Theorem of Galois Theory. Applications: solvability of
algebraic equations - The fundamental theorem of Algebra - Roots of unity.
Course Bibliography (Eudoxus)
1. Εισαγωγή στην Άλγεβρα, J. Fraleigh, ISBN: 978-960-7309-71-6, Πανεπιστημιακές Εκδόσεις Κρήτης
2. Άλγεβρα, Δ. Μ. Πουλάκης, ISBN 978-960-456-388-3, Εκδόσεις ΖΗΤΗ, Θεσσαλονίκη, 2015
3. Θεωρία Galois, J. Rotman, ISBN: 9607901126, ΔΙΑΔΡΟΜΕΣ
Additional bibliography for study
Θεωρία Galois, Θ. Θεοχάρη-Αποστολίδη, Χ. Χαραλάμπους, ISBN: 978-960-603-208-0, [ηλεκτρ. βιβλ.] Αθήνα:Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, ID Ευδόξου: 320037