A simple icon, consisting of a series of links surrounded by a circle, is a common symbol for a connection or hyperlink. The lines of the icon are outlined in a dark black color on the background, with the addition of military elements of the Saudi national identity such as the Ghutra, Shamaa, and Saudi Bisht, to reflect the distinctive local character of Qassim University.

Links to official Saudi educational websites end with

edu.sa

All links to official educational websites of government agencies in Saudi Arabia end with .edu.sa.

Black leather minimalist gesture tag, black circular grip, topped with a clear depiction of a Saudi tunic with a shamma and aqal, emphasizing the features of the Saudi bisht. This design symbolizes the concept of security and digital data privacy and reflects the identity of Qassim University.
protocol for encryption and security. HTTPS for encryption and security.
Secure websites in the Kingdom of Saudi Arabia use the HTTPS protocol for encryption.
Digital Government Authority
Registered with the Digital Government Authority with no: 20260416249
Qassim University logo
Open search engine
Search
MyQU Single Sign-On Portal
MyQU

Colleges

Section list

Loops and fields

Course Description: 1. Rings and examples thereof. 2. Ring homomorphisms and isomorphisms and their properties. 3. Properties of integral domains, division rings (skew fields), and fields. 4. Prime fields, subrings, subfields, and the characteristic of a ring. 5. Polynomial rings. 6. Properties of polynomial rings. 7. Greatest common divisor of polynomials and the division algorithm for polynomials. 8. Reducible and irreducible polynomials. 9. Irreducible polynomials over the integers and rational numbers. 10. Ideals and their properties. 11. Generated ideals and the direct sum of ideals. 12. Quotient rings and some examples thereof. 13. First, second, and third isomorphism theorems. 14. Principal ideals, prime ideals, and maximal ideals. 15. Field extensions.
Credit hours: 4
Prerequisites: MATH.343
Objectives of the course :

The course topics, learning outcomes, and teaching strategies revolve around enabling the student to do the following:
• Definition of the concepts of rings and field theory starting from the concept of group theory.
• Formulating and proving numerous theorems by studying the mathematical background of number theory, linear algebra, and group theory.
• Analytically re-presenting aspects of solving reducible (and irreducible) polynomials by explaining the mathematical theory underlying the solution methods.
• Comparing rings and fields in terms of their properties, in addition to the ability to solve problems related to polynomials and types of ideals by solving problems related to these concepts.
• Applying different methods to solve practical problems related to this course.
• Differentiating between types of ideals by solving various examples.
• Calculating the direct sum, product, and sum of ideals, and examples of polynomial division operations, by solving numerous practical examples through collaborative thinking.
• Justify the polynomial division algorithm by working within a team.

Course outputs :

The course covers the following topics:
Rings and examples of them, ring homomorphisms and isomorphisms, and their properties.
Properties of integral domains, fields, prime fields, subrings, subfields, and the characteristic of a ring.
Polynomial ring.
Properties of the polynomial ring, greatest common divisor of polynomials, and the polynomial division algorithm.
Reducible and irreducible polynomials.
Irreducible polynomials over the integers and rational numbers.
Ideals and their properties, generated ideals, and the direct sum of ideals.
Quotient rings (factor rings) and some examples of them.
The first, second, and third isomorphism theorems.
Principal ideal, prime ideal, and maximal ideal.

Additional information:
  • Last update: 2026-09-15

Suggestions

Single Sign-On Portal
Achievement
E-learning
Email
Directory of services
Query services

Cookies

This website uses special cookies to ensure ease of use, improve your browsing experience, and clarify the terms and policies related to About user privacy. By continuing to browse this website, you acknowledge that you accept the use of cookies and the terms of the Privacy Policy

Accept