# Introduction To Rings And Fields

By Prof. Krishna Hanumanthu   |   Chennai Matematical Institute
Learners enrolled: 1355
This course will cover basics of abstract rings and fields, which are an important part of any abstract algebra course sequence. We will spend roughly the 4-5 weeks on rings. We will begin with definitions and important examples. We will focus cover prime, maximal ideals and important classes of rings like integral domains, UFDs and PIDs. We will also prove the Hilbert basis theorem about noetherian rings. The last 3-4 weeks will be devoted to field theory. We will give definitions, basic examples. Then we discuss extension of fields, adjoining roots, and prove the primitive element theorem. Finally we will classify finite fields.

INTENDED AUDIENCE: B.Sc and M.Sc students studying mathematics
PREREQUISITES: A little bit of abstract group theory and a little bit of linear algebra.
INDUSTRY SUPPORT: None
Summary
 Course Status : Completed Course Type : Core Duration : 8 weeks Category : Mathematics Foundations of Mathematics Algebra Credit Points : 2 Level : Postgraduate Start Date : 24 Jan 2022 End Date : 18 Mar 2022 Enrollment Ends : 07 Feb 2022 Exam Date : 27 Mar 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week 1: Definition of rings, examples, polynomial rings, homomorphisms.
Week 2: Ideals, prime and maximal ideals, quotient rings.
Week 3: Noetherian rings, Hilbert basis theorem.
Week 4: Integral domains, quotient fields.
Week 5: Unique factorization domains, principal ideal domains.
Week 6: Definition of fields, examples, degree of field extensions.
Week 7: Adjoining roots, primitive element theorem.
Week 8: Finite fields.

### Books and references

Algebra by Michael Artin

### Prof. Krishna Hanumanthu

Chennai Matematical Institute
Krishna Hanumanthu is an associate professor of mathematics at Chennai Mathematical Institute (CMI). He studied BSc and MSc in CMI during 1998-2003 and did his PhD in mathematics at University of Missouri during 2003-2008. He joined CMI as a faculty member in 2011 after working for 3 years at University of Kansas. His main areas of research are algebraic geometry and commutative algebra. He has been teaching for almost 15 years and taught introductory courses on abstract algebra (including group theory) many times.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 27 March 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 6 assignments out of the total 8 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team