Field theory (mathematics)
Lewis, D. J. Department of Mathematics, University of Michigan, Ann Arbor, Michigan.
- Subfields and extensions
- Solutions of polynomial equations
- Automorphisms of fields
- Links to Primary Literature
- Additional Readings
In algebra, the term field is used to designate an algebraic system or structure containing at least two elements and having two binary rules of composition: addition and multiplication (that is, if a and b are any two elements of the field, then a + b and ab are defined and are elements of the field). The structure rules are as follows: The elements form an abelian (commutative) group under addition with the additive identity denoted by 0; that is, a + 0 = a for all elements a. The set of nonzero elements (and there are some since the field has at least two elements) form an abelian group under multiplication with the multiplicative identity denoted by 1. It follows that all nonzero elements have a multiplicative inverse or reciprocal. The two rules of composition are related by the distributive law: (a + b)c = ac + bc for all elements a, b, c. It follows from the distributive law that a · 0 = 0 for all elements a, since 1 · a = (1 + 0)a = 1 · a + 0 · a, whence 0 = 0 · a. See also: Group theory
The content above is only an excerpt.
for your institution. Subscribe
To learn more about subscribing to AccessScience, or to request a no-risk trial of this award-winning scientific reference for your institution, fill in your information and a member of our Sales Team will contact you as soon as possible.
to your librarian. Recommend
Let your librarian know about the award-winning gateway to the most trustworthy and accurate scientific information.
AccessScience provides the most accurate and trustworthy scientific information available.
Recognized as an award-winning gateway to scientific knowledge, AccessScience is an amazing online resource that contains high-quality reference material written specifically for students. Its dedicated editorial team is led by Sagan Award winner John Rennie. Contributors include more than 9000 highly qualified scientists and 39 Nobel Prize winners.
MORE THAN 8500 articles and Research Reviews covering all major scientific disciplines and encompassing the McGraw-Hill Encyclopedia of Science & Technology and McGraw-Hill Yearbook of Science & Technology
115,000-PLUS definitions from the McGraw-Hill Dictionary of Scientific and Technical Terms
3000 biographies of notable scientific figures
MORE THAN 17,000 downloadable images and animations illustrating key topics
ENGAGING VIDEOS highlighting the life and work of award-winning scientists
SUGGESTIONS FOR FURTHER STUDY and additional readings to guide students to deeper understanding and research
LINKS TO CITABLE LITERATURE help students expand their knowledge using primary sources of information