ALGEBRA I
cod. 05544

Academic year 2007/08
1° year of course - Second semester
Professor
Academic discipline
Algebra (MAT/02)
Field
Formazione algebrico-geometrica
Type of training activity
Characterising
60 hours
of face-to-face activities
9 credits
hub:
course unit
in - - -

Learning objectives

<br />Demonstrate the existence of some algebraic theories also providing basic nomenclature and elementary information within a universal context.
<br />Students should also show interpretation and orientation capabilities.

Prerequisites

Course in Linear Algebra

Course unit content

<br />1. BASIC PRINCIPLES AND EXAMPLES. EXTERNAL AND INTERNAL OPERATIONS. GROUPOIDS, SUB-GROUPOIDS, DIRECT PRODUCTS, HOMOMORPHISMS, ISOMORPHISMS. CONVOLUTIONS, POLYNOMIALS, NUMBERS (COMPLEX, DUAL, BI REAL; GAUSS WHOLE NUMBERS) AND MATRICES. BASIC THEOREMS ON HOMOMORPHISMS, CONGRUENCES, QUOTIENTS, NORMAL SUBGROUPS, IDEALS. LATERALS AND THEOREM OF LAGRANGE. POLYNOMIALS AND POLYNOMIAL FUNCTIONS. <br />
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2. ARITHMETIC QUESTIONS AND APPLICATIONS. LOCALISATION. DOMAINS. EUCLIDEAN DIVISION, EUCLIDEAN RINGS, MAIN IDEALS. CYCLICAL GROUPS, CHARACTERISTICS OF A RING, PRIME FIELDS. QUADRATIC DOMAINS. IRREDUCIBLE AND PRIME ELEMENTS. EXTENDED EUCLIDEAN ALGORITHM AND BEZOUT’S FORMULA. FACTORISATION THEOREM. DETAILS OF Z METHODOLOGY: SECOND DEGREE EQUATIONS. POLYNOMIAL EQUATIONS THEOREM OF RUFFINI. FIELD EXTENSIONS. FIELD SIZE. ALGEBRAICALLY CLOSED FIELDS. FINITE FIELDS.  

Full programme

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Bibliography

No text book has been adopted but an extensive list of classic texts, printouts from courses and notes downloadable free of charge from the Internet has been provided.

Teaching methods

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Assessment methods and criteria

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Other information

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