Dokuz Eylül University Information Package / Courses Catalog

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type Of Course	D	U	L	ECTS
MAT 4065	COMPUTATIONAL COMMUTATIVE ALGEBRA I	ELECTIVE	4	0	0	7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

PROFESSOR DOCTOR GÖKHAN BILHAN

Offered to

Mathematics (Evening)
Mathematics

Course Objective

The aim of this course is to introduce methods of commutative algebra with a view towards algebraic geometry and with the computational methods using Groebner basis.

Learning Outcomes of the Course Unit

1	Will be able to find Groebner basis of a given ideal in the polynomial ring over a field with n indeterminates.
2	Will be able to determine whether a given polynomial is in a given ideal in the polynomial ring over a field with n indeterminates using Groebner basis and division algorithm.
3	Will be able to solve a given system of polynomial equations over a field with n indeterminates under suitable conditions using the Elimination Theorem and the Extension Theorem.
4	Will be able to find the smallest variety containing a set defined by polynomial parametrizations by using the Elimination Theorem.
5	Will be able to interpret the relation between ideals and varieties using the ideal-variety correspondence and the Hilbert s Nullstellensatz when the field is algebraically closed.
6	Will be able to interpret the coordinate ring of an affine variety as a quotient ring of the polynomial ring.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week	Subject	Description
1	Affine varieties. Parametrizations of affine varieties. Ideals.
2	Monomial orders. Division algorithm. Dickson's Lemma for monomial ideals. The Hilbert Basis Theorem.
3	Groebner bases. Buchberger's Algorithm.
4	Applications of Groebner bases.
5	Elimination and Extension Theorems.
6	The geometry of eliminination. Implicitization.
7	Singular points and envelopes.
8	Resultants and the Extension Theorem.
9	Midterm
10	Hilbert's Nullstellensatz. Radical ideals and the Ideal-Variety Correspondence, the algebra-geometry dictionary.
11	Sums, product and intersections of ideals. Zariski closure and quotients of ideals.
12	Irreducible varieties and prime ideals. Decomposition of a variety into irreducible varieties. Primary decomposition of ideals.
13	Polynomial mappings. Quotients of polynomial rings.
14	The coordinate ring of an affine variety. Rational functions on a variety. Closure Theorem.

Recomended or Required Reading

Textbook(s): Cox, D., Little, J. and OShea D. Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, Springer, 2007.
Supplementary Book(s): 1) Reid, M. Undergraduate Algebraic Geometry, Cambridge University Press, 1998.
2) Reid, M. Undergraduate Commutative Algebra, Cambridge University Press, 1995.
3) Greuel, G.-M. and Pfister, G. A Singular Introduction to Commutative Algebra, With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, Second, Extended Edition, Springer, 2008.
References: 1) Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Springer, 1995.
2) Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra, Addison Wesley, 1994.
3) Sharp, R. Y. Steps in Commutative Algebra, Second edition, Cambridge University Press, 2004.
4) Matsumura, H. Commutative Ring Theory, Cambridge University Press, 1989.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving, Discussion

Assessment Methods

SORTING NUMBER	SHORT CODE	LONG CODE	FORMULA
1	MTE	MIDTERM EXAM
2	FIN	FINAL EXAM
3	FCG	FINAL COURSE GRADE	MTE * 0.40 + FIN * 0.60
4	RST	RESIT
5	FCGR	FINAL COURSE GRADE (RESIT)	MTE * 0.40 + FIN * 0.60

*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Lectures	13	4	52
Preparations before/after weekly lectures	12	4	48
Preparation for midterm exam	1	30	30
Preparation for final exam	1	30	30
Final	1	3	3
Midterm	1	3	3
TOTAL WORKLOAD (hours)			166

Contribution of Learning Outcomes to Programme Outcomes

PO/LO	PO.1	PO.2	PO.3	PO.6	PO.8	PO.10
LO.1	5	3	5	3	3	3
LO.2	5	3	5	3	3	3
LO.3	5	3	5	3	3	3
LO.4	5	3	5	3	3	3
LO.5	5	3	5	3	3	3
LO.6	5	3	5	3	3	3

COURSE UNIT TITLE

DEGREE PROGRAMMES

Description of Individual Course Units

Offered By

Level of Course Unit

Course Coordinator

Offered to

Course Objective

Learning Outcomes of the Course Unit

Mode of Delivery

Prerequisites and Co-requisites

Recomended Optional Programme Components

Course Contents

Recomended or Required Reading

Planned Learning Activities and Teaching Methods

Assessment Methods

Further Notes About Assessment Methods

Assessment Criteria

Language of Instruction

Course Policies and Rules

Contact Details for the Lecturer(s)

Office Hours

Work Placement(s)

Workload Calculation

Contribution of Learning Outcomes to Programme Outcomes

CONTACT INFORMATION