This is an audio version of the Wikipedia Article:
00:01:55 1 Definition
00:02:42 1.1 Classic definition
00:02:46 1.2 Alternative definition
00:03:39 2 Examples
00:03:56 2.1 Rational numbers
00:04:15 2.2 Real and complex numbers
00:04:56 2.3 Constructible numbers
00:05:20 2.4 A field with four elements
00:05:30 3 Elementary notions
00:06:27 3.1 Consequences of the definition
00:06:31 3.2 The additive and the multiplicative group of a field
00:06:41 3.3 Characteristic
00:10:35 3.4 Subfields and prime fields
00:11:19 4 Finite fields
00:12:23 5 History
00:14:09 6 Constructing fields
00:14:52 6.1 Constructing fields from rings
00:15:04 6.1.1 Field of fractions
00:15:52 6.1.2 Residue fields
00:16:08 6.2 Constructing fields within a bigger field
00:16:19 6.3 Field extensions
00:16:45 6.3.1 Algebraic extensions
00:16:56 6.3.2 Transcendence bases
00:17:11 6.4 Closure operations
00:18:41 7 Fields with additional structure
00:20:27 7.1 Ordered fields
00:21:11 7.2 Topological fields
00:22:04 7.2.1 Local fields
00:22:50 7.3 Differential fields
00:23:26 8 Galois theory
00:24:10 9 Invariants of fields
00:24:34 9.1 Model theory of fields
00:24:50 9.2 The absolute Galois group
00:25:37 9.3 K-theory
00:26:54 10 Applications
00:30:06 10.1 Linear algebra and commutative algebra
00:30:16 10.2 Finite fields: cryptography and coding theory
00:31:33 10.3 Geometry: field of functions
00:33:50 10.4 Number theory: global fields
00:34:50 11 Related notions
00:35:41 11.1 Division rings
00:36:42 12 Notes
00:37:17 13 References
00:37:34 Algebraic extensions
00:40:09 Transcendence bases
00:41:31 Closure operations
00:43:51 Fields with additional structure
00:44:11 Ordered fields
NaN:NaN:NaN Topological fields
NaN:NaN:NaN Local fields
NaN:NaN:NaN Differential fields
NaN:NaN:NaN Galois theory
NaN:NaN:NaN X5 − 4X + 2 (and E
NaN:NaN:NaN Invariants of fields
NaN:NaN:NaN Model theory of fields
NaN:NaN:NaN The absolute Galois group
NaN:NaN:NaN H2(F, Gm).
NaN:NaN:NaN F×. Matsumoto's theorem shows that K2(F) agrees with K2M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.
NaN:NaN:NaN === Linear algebra and commutative algebra
NaN:NaN:NaN bhas a unique solution x in F, namely x
NaN:NaN:NaN Finite fields: cryptography and coding theory
NaN:NaN:NaN x3 + ax + b.Finite fields are also used in coding theory and combinatorics.
NaN:NaN:NaN Geometry: field of functions
NaN:NaN:NaN Number theory: global fields
NaN:NaN:NaN Related notions
NaN:NaN:NaN Division rings
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In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle can not be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.
Fields serve as foundational notion ...