Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Field (mathematics)

3 Some first theorems

• The set of non-zero elements of a field F (typically denoted by F^×) is an abelian group under multiplication. Every finite subgroup of F^× is cyclic.

• The characteristic of any field is zero or a prime number. (The characteristic is defined as the smallest positive integer n such that n·1 = 0, or zero if no such n exists; here n·1 stands for n summands 1 + 1 + 1 + ... + 1.)

• The number of elements in finite fields is a prime power.

• As a ring, a field has no ideals except {0} and itself.

• For every field F, there exists a (up to isomorphism) unique field G which contains F, is algebraic over F, and is algebraically closed. G is called the algebraic closure or F.

4 Constructing new fields from given ones

1. If a subset E of a field (F,+,*) together with the operations *,+ restricted to E is itself a field, then it is called a subfield of F. Such a subfield has the same 0 and 1 as F.
2. The polynomial field F(x) is the field of fractions of polynomials in x with coefficients in F.
3. An algebraic extension of a field F is the smallest field containing F and a root of an irreducible polynomial p(x) in F[x]. Alternatively, it is identical to the factor ring F[x]/<p(x)>, where <p(x)> is the ideal generated by p(x).

5 See also

• See field theory (mathematics) for some history and other information.
• See Glossary of field theory for more definitions in field theory.
• Fields - wikibook link.

Politics	Business	Finance

Topics: Field (mathematics) Numbers Fields *the Set Theory Algebraic...

---

Field (mathematics)In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil Fields MedalThe Fields Medal is a prize awarded to up to four mathematicians (not over forty years of age) at each International Congress of International Mathematical Union, since 1936 and regularly since 1948 at the initiative of the Canadian mathematican John Char Field hockeyField hockey is a popular sport for men and women in many countries around the world. It is simply known as hockey in most countries, especially those in which ice hockey is not very prominent. Field hockey has several regular, prestigious international t Field-programmable gate arrayA field-programmable gate array or FPGA is a gate array that can be reprogrammed after it is manufactured, rather than having its programming fixed during the manufacturing — a programmable logic device. FPGAs are generally slower than their ASIC counterp Field effect transistorThe Field-Effect Transistor (FET) is a family of transistors that rely on an electric field to control the conductivity of a "channel" in a semiconductor material. FETs, like all transistors, can be thought of as voltage-controlled resistors. Most FETs ar Field extensionIn abstract algebra, an extension of a field K is a field L which contains K as a subfield. For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers). Field ion microscope Field strength Fielding (cricket) Field gun Field camera Field Emission Electric Propulsion Fieldon, Illinois Field Township, Minnesota Fieldon Township, Minnesota