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. F

    Engineering Calculating the charge if the electric field density = 0

    Question: Relevant Equations: My attempt: Could someone please confirm my solution?
  2. Z

    On which objects can we apply Gauss' Law to find the electric field?

    What am I missing? I also don't get the title of the section: "Charge distributions with enough symmetry for Gauss's Law". I thought Gauss's Law was valid for any closed surface enclosing a charge. I don't understand what "enough symmetry" means in the title above. I get that with symmetry...
  3. Z

    Electric field is constant around charged infinite plane. Why?

    Using Gauss's Law By using a symmetry argument, we expect the magnitude of the electric field to be constant on planes parallel to the non-conducting plane. We need to choose a Gaussian surface. A straightforward one is a cylinder, ie a "Gaussian pillbox". The charge enclosed is...
  4. P

    A Einstein Field Equations: Spherical Symmetry Solution

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  5. godiswatching_

    What is the effect of a 1 cm gap on the magnetic field in a solenoid?

    Hey, I was trying to figure out this problem. I got (a) using B = mu * NI/L but I'm not sure how to start the part about the magnetic field in the gap after the solenoid is ripped in half with 1 cm gap. Thanks for the help!
  6. F

    I Can a static EM field consist of photons?

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  7. P

    A How to read modern condensed matter field research

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  8. Boltzman Oscillation

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  9. Z

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  10. Z

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  11. C

    I Average field inside spherical shell of charge

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  12. guyvsdcsniper

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  13. H

    I How does QM explain that we see electrons circulating in a magnetic field?

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  14. H

    I Image for increase in gravitational potential energy in radial field

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  15. F

    I Electric field of uniformly polarized cylinder

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  16. desperatestudent123

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  17. Hamiltonian

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  18. Dezzi

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  19. BadgerBadger92

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  20. R

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  21. A

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  22. LCSphysicist

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  23. J

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  24. Tertius

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  25. J

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  26. J

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  27. patric44

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  28. A

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  29. Caleydos

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  30. mohammed

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  31. tworitdash

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  32. H

    Experiment Exposing Tissue to Pulsed Electromagnetic Field

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  33. F

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  34. Ethan

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  35. stephenklein

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  36. Ugnius

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    So I started with b) and it there was no q2 this would seem reasonable I was wanted to ask , what effect does q2 have on potential of these two charges? Because it has to be given for a reason.
  37. Shreya

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  38. Sciencemaster

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  39. A

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  40. J

    Is the magnetic field B→. a state function and exact differential?

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  41. M

    I Gravitational Field of Electromagnetic Waves: How is it Generated?

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  42. guyvsdcsniper

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  43. guyvsdcsniper

    Help finding the Electric field at the center of charged arc

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  44. Ugnius

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  45. Sagittarius A-Star

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  46. karush

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  47. Tertius

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  48. V

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  49. bob012345

    I Divergence of the Electric field of a charged circular ring

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  50. Noki Lee

    I Investigating the 1d Equation: Charges & Field Disparity

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