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. arivero

    I Is Connes' Model of the Higgs Field a Wormhole?

    As this week the great topic is the Einstein-Rosen bridge and duals of it, I wonder what happened with the interpretation of Connes NCG model as two sheets of spacetime whose separation is measured by the higgs field. How is it different of a ER bridge?
  2. C

    Electric Field of a Uniform Ring of Charge

    Hi! For this problem, The solution is, However, why did they not include constants of integration in their working shown in red? Many thanks!
  3. alan123hk

    B Decompose the E field into conservative and non-conservative parts

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  4. livio

    Estimates of voltage drop with distance in weak electric field

    If I resolve the equation in 0, imposing a voltage value of 5 mV, it gives a non real solution, therefore I cannot resolve it for R=1 because I do not know which voltage value to impose. I am sure this is simpler than I am putting it :) thanks for any advice!
  5. C

    Why Does Positive Charge Exert Elec. Field Beyond Neg. Charge?

    I wonder how it is possible that a positive charge can exert el. field beyond negative charge? Shouldn't they "connect" and therefore positive charge should stop to have el. field beyond neg. charge? I mean, I am obviously wrong about that, but can someone please explain why/how el. field from...
  6. ermia

    Finding Constants: Potential and Field Analysis

    I have wrote all feilds and potentials and I want to find the constants. My first question is " when we say in the a<x<2a the potential is V(x)" then the potential in the a is V(a) or V(0) ( cause it is 0 in our new area) ? Second one is " when I want to write the gausses law for the point x=a I...
  7. M

    A The Effect of Time-Varying Electric Fields on a 2-Level Ion System

    Hello! If I have an ion which can be treated as a 2 level system, in a time varying electric field (the variation of the field doesn't need to be on or close to resonance, but for simplicity we can assume it is an oscillatory field) can I simply separate the problem into a center of mass motion...
  8. redirmigician

    I Acceleration in an electron field

    I read something about accelerators using nanotubes. I am a little concerned about the design mentioned in the "High Density with Perpendicular Carbon Nanotubes" part of this paper(https://doi.org/10.3390/photonics8060216). Can wakefield acceleration be done in an electron field? Or maybe I...
  9. A. Neumaier

    A Ensembles in quantum field theory

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  10. cwill53

    Average Electric Field over a Spherical Surface

    The picture above shows the integral that needs to be evaluated, and the associated picture ## \cos\alpha ## can be obtained via the law of cosines. I'm simply confused as to where the ##\cos\alpha ## comes from in the first place. I just don't see why ##\cos\alpha ## is necessary in this...
  11. Green dwarf

    I Time Dilation in Gravitational Field: Potential vs Field Strength

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  12. N

    Find the Electric field at point p

    Since q3=q4 and they are opposite to each others they cancel out But as soon as I try to find the electric field of one of the charges, I need the radius which is not given. By isolating the electric field for radius E=(kq)/r^2 I now have two unknowns
  13. Povel

    A Exploring the Electric Field of a Moving Charged Spherical Shell

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  14. Y

    Electric field of charged rod

    hello i would like to understand to something. here is the drew now for my question: i was able to find Ey and here is my correct answer: when i try to find Ex i didnt understand something, i found the correct answer but i need to put minus before and i want to know why? here is my solution...
  15. H

    I Why is the electric field in y direction in a TE guided wave?

    Hi, I have a fairly simple question, but the answer is probably not as simple. I'm not sure to understand why in a guided wave (TE), the electric field is in the y direction. I know ##E_z = 0##, but why ##E_x = 0, B_y = 0##?
  16. S

    I How do I visualize the magnetic field?

    Hi! So my question is this, I have done measurements with an magnetic field meter around a transformer from 0.5 meter away (then measure some points around) and then I moved out 0.5 meters and so on until I reached a nearby building. So my issue now is I want to visualize this to my customer...
  17. V

    Why does a diamagnetic rod align perpendicular to a magnetic field?

    I know that each material is made up of tiny magnets due to electrons orbiting the nucleus and also from electron spinning about its own axis. In ferromagnetic or paramagnetic rod these tiny magnets align with the applied field causing the net field in the rod to increase. But for diamagnetic...
  18. T

    I How is an interference pattern changed by an external field?

    For example, through the diffracting of electrons and their wave-particle dualistic nature, its possible to produce an interference pattern. And for the resulting electron beam, it will still be affected by the Lorentz force and be deflected by it. At the same time, the position(s) of the...
  19. J

    A The Probability Distribution of a Bosonic Field when Emitted

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  20. N

    I Questions about a Conductor in an Electric Field

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  21. C

    I Where do discontinuities in the electromagnetic field occur?

    Obviously at point charges, but where along boundaries? Would they theoretically occur in superconductors since they can carry infinite current (J -> infinity)?
  22. G

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  23. T

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  24. gentzen

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  25. Barbequeman

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  26. guyvsdcsniper

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  27. S

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  28. H

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  29. R

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  30. P

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  31. Fra

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  32. H

    Why Does the Electric Field Sum Instead of Cancel with Opposite Charges?

    If there are two charges positive and negative and their electric field point in the same direction then the total electric field would be their sum of magnitudes. Why don't we consider the sign of the charges? For example, a parallel plate capacitor is inside the region where both the positive...
  33. J

    I Field fluctuations in the vacuum

    How does relativistic qft predict quantum fluctuations in the vacuum? We see this in the experiment proving the Casimir Effect so we know it's physical, but why?
  34. P

    Evaluating the Integral of a Vector Field Using Cauchy-Schwarz Inequality

    Here is my attempt (Note: ## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|## ##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ## Here I am going to surround the above expression with another set of...
  35. lindberg

    I Unruh, Haag et al.: No Room for Particles in Quantum Field Theory?

    In a paper by Bain (2011), particles are left with little ontological value because of the Reeh-Schlieder theorem, the Unruh effect and Haag's theorem. The author claims (and here I am copying his conclusion): First, the existence of local number operators requires the absolute temporal metric...
  36. ermia

    Electric field needed to tear a conducting sphere

    ..
  37. lindberg

    I Haag's Theorem: Explain Free Field Nature

    What is the main reason for a free field staying free according to Haag's theorem?
  38. ermia

    Electric field of a part of a hemisphere

    I tried gauss law. And the fact that if alpha is less than pi/2 we can say that we have two parts with angle alpha and one other part which has a normal field at the center. But non of them helped me answer. The problem's solution says that we can use the fact that our section has longitudinal...
  39. Rikudo

    Gravitational field of a hollow sphere

    Why the area of the thin rings are ##2πasin\theta \, ds##? (a is the radius of the hollow sphere) If we look from a little bit different way, the ring can be viewed as a thin trapezoid that has the same base length ( ##2πa sin\theta##), and the legs are ## ds##. The angle between the leg and...
  40. B

    EMF Induced by Changing Magnetic Field

    Can anyone show me where I'm wrong?
  41. C

    I Do objects of differing mass fall at the same rate in a magnetic field?

    Gravity isn't a force in the strictest sense of the word, yet magnetism is exactly that: a force. As is strong, EW, etc. Therefore, it's possible that the more massive magnetic object gets drawn to the center of a magnetic source at a faster rate than the less massive magnetic object. Discuss!
  42. thedubdude

    I Why does metal moving though a magnetic field slow down?

    A piece of metal moving West to East in a North to South fixed magnetic field slows down...but how? Yes of course eddy currents are set up in the metal and these currents generate their own magnetic field which somehow slows down the moving metal piece...but how does this actually slow the...
  43. besebenomo

    Magnetic flux with magnetic field changing direction

    Sorry if I post again about this topic (last time I promise!) but I still have some doubts regarding the concept of flux. This collection of problems I have quite standard but there are so many variations. Here is the circuit in question: Something tells me that I could write a function that...
  44. warrenchu000

    Why is magnetic field B along a straight wire circular not radial?

    Statement: The magnetic field around a straight wire carrying a current can be explained Relativistically by changing the inertial frame of reference to the frame of the moving electrons - i.e., a Lorentz contraction of the positive charges in the wire will give a denser concentration of the...
  45. besebenomo

    Moving bar enclosing a changing magnetic field generates a current

    The amplitude of ##\vec{B}## is given by: $$B(x) = B_{0} - B_{0} \frac{x}{2l}$$ for ##0 \leq 0 \leq 2l## This was my attempts at finding the flux of B: $$\Phi(B) = (B_{0} - B_{0} \frac{x}{2l})(2l-x(t))l = B_{0}2l^2-2B_{0}x(t)l+ B_{0}\frac{x(t)^2}{2}$$ and the current: $$ i = -\frac{d...
  46. besebenomo

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    $$B(t) = B_{0} \frac{t^2}{T^2}$$ for ##0 \leq t \leq T## The issue here is more conceptual, because once I find the flux of B I know how to proceed to find the current. I got velocity (but it seems to me that it is the initial velocity), I could use it to find the time in function of space...
  47. Ahmed1029

    I How do I find the Direction of an induced electric field?

    Faraday's law tell's you about the line intergal of the electric field, but you have to know the direction of the induced electric field first in order to properly apply it. How can I find its direction? Is it in the same direction as the induced current?
  48. G

    B The relationship between the particle, the wave and the field

    What is it the we detect in the first instance? Is it the particle |wave or is it the field? Is the former more fundamental than the latter in any sense or are we just talking the opposite sides of the same coin? For instance does the em field create the photon and the electron or could...
  49. warhammer

    B Direction of Electric Field & Field Due to a Dipole

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  50. FFXT

    I Faraday induction in constant B field, with non-conduction wires

    A standard textbook problem features a constant B field and a conducting loop that increases in area at constant rate. It is easy to work out the induced EMF and the associated electric field magnitude and direction (CW or CCW). The magnitude of the E field is E = B v where v is a velocity...
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