# Matematica | GALOIS THEORY

## Matematica GALOIS THEORY

 0512300021 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2013/2014

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 GIOVANNI VINCENZI T
Objectives
THE GOAL IS TO STUDY THE ALGEBRAIC EQUATIONS. FOLLOWING GALOIS'S THEORY, IT WILL BE CHARACTERIZED THE EQUATIONS THAT ARE SOLUBLE BY RADICALS. CONNECTIONS WITH OTHER AREAS OF MATH WILL BE HIGHLIGHTED.
Prerequisites
FULL KNOWLEDGE OF ELEMENTARY ALGEBRA IS REQUIRED
Contents
IRRIDUCIBLITY OF A POLINOMIAL. EXTENDED ISOMORPHISM THEOREMS. SPLITTING FIELD OFA POLINOMIAL. ORDINALAND CARDINAL NUMBERS: TRASFINITY INDUCTION. ALGBRICALLY CLOSED FIELDS, AND ALGEBRIC CLOSURE OF A FIELD.
CYCLOTOMIC POLINOMIALS, THEOREM OF GAUSS. NORMAL AND SEPARABLE EXTENSIONS. THE CARDANO'S FORMULAS FOR DETERMINE THE ROOTS OF AN EQUATION OF THIRD OR FOURTH DEGREE. GALOIS EXTENSIONS AND THEIR CARACTERIZATION
GALOIS APPLICATIONS: FOUNDAMENTAL THEOREM OF GALOIS THEORY. RISOLUBILITY BY RADICALS OF A GIVEN EQUATION,AND THEIR CARACTERIZATION.
CONSTRUCTIONS BY STRAIGHTEDGE AND COMPASS.
Teaching Methods
USUAL
Verification of learning
THERE ARE MORE ELEMENT THAT GIVE CONTRIBUTION FOR THE VALUTATION OF A STUDENT.
1) INTERACTION DURING LECTURES
2) CHECK AND DISCUSSION OF THE HOMEWORKS
3) WRITING EXAMS
4) ORAL EXAM
Texts
1)ARTIN, ALGEBRA, BOLLATI-BORINGHIERI
2) NOTES GIVEN DURING THE LESSONS
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