What is Field: Definition and 1000 Discussions

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 cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

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  1. K

    I Are gravitons necessary?

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  2. hongseok

    B How is a magnetic field formed in a space surrounded by one pole?

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  3. J

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  4. tellmesomething

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  5. tellmesomething

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  6. ergospherical

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  7. kated

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  8. M

    B Solar Wind Particles, - reaching into Earth's magnetic field?

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  9. aryan kumar pandey

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  10. Ben2

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  11. maxelcat

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  12. Rick16

    B What ever happened to Classical Unified Field Theory?

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  13. L

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  14. B

    Planar Coil Rotating in a Magnetic Field

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  15. Hill

    A Vector field vs. four scalar fields ("QFT and the SM", Schwartz)

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  16. Hill

    I What does "transforms covariantly" mean here?

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  17. A

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  18. K

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  19. tanmays1

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  20. R

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  21. Hill

    Quantum 'Quantum Field Theory for the Gifted Amateur' by Tom Lancaster and Stephen J. Blundell as a self-study textbook?

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  22. Slimy0233

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  23. paulimerci

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  24. paulimerci

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  25. T

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  26. carlsondesign

    I What is the official name for a Field Series in mathematics/physics?

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  27. harryharns

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  28. ergospherical

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  29. C

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  30. D

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  31. StenEdeback

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  32. R

    B Exploring the Electric Field of a Moving Charge

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  33. I

    I Exploring Field Strength & Singularities

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  34. M

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  35. CaliforniaRoll88

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  36. ahmadphy

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  37. milkism

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  38. R

    B Seeing both B field lines and E field lines at the same time

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  39. M

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  40. S

    I Gravitational field modulation of a moving mass

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  41. R

    A Bound states of an electron trapped in a dipole field

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  42. R

    A Aligning effect in uniform field

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  43. L

    Divergence of the Electric field of a point charge

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  44. N

    I Electric Field Shielding by Conducting Sheets

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  45. milkism

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  46. O

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  47. simonjech

    I Necessity of the field viewpoint

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  48. S

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  49. K

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  50. S

    I Propagator of massless Weyl field

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