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. N

    Relativistic motion of an electron in a uniform electric field

    dv/dt is the acceleration, so I thought I could find the acceleration from F = qE = ma = dp/dt. But this is a relativistic case, so the proper acceleration is a = F/mγ3, where v in the gamma is the v of the electron and F = eE. However, I'm not sure if this is correct, because the constant τ...
  2. Shacking

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  3. Adesh

    Can a magnetic field ever cause a translation motion?

    Lorentz Law says that for a charged particle moving with a velocity v in a magnetic field B then the force on it is given by $$ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) $$ Now, if I say that particle’s velocity and the magnetic field are aligned then according to Lorentz Law there will be...
  4. Luke Tan

    On-Axis Field of a Uniformly Magnetized Sphere

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  5. PrimoCosta

    Eletromagnétic field representation

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  6. N

    Electric Field of a Point Charge

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  7. Math Amateur

    MHB Operator norm .... Field, Theorem 9.2.9 ....

    I am reading Michael Field's book: "Essential Real Analysis" ... ... I am currently reading Chapter 9: Differential Calculus in \mathbb{R}^m and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ... I need some help in fully understanding Theorem 9.2.9 (3) ...
  8. patrykh18

    Magnetic field outside a solenoid

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  9. BabyHueyTAW

    Definition: What is meant by port "CARP" ASIC functionality in a Field Programmable Gate Array?

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  10. Math Amateur

    MHB Operator Norm .... differences between Browder and Field ....

    I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the differences between Andrew Browder and Michael...
  11. D

    How to find the magnetic field and magnetic force due to a solenoid loop

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  12. BickySmalls

    What would happen to a charge that is placed in an Electric Field = 0?

    If a charge was placed inside an electric field, where the electric field was equal to zero, what effect would the charge have?
  13. M

    Electromagnetics question - magnetic field and electrolysis

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  14. E

    B Change in field strength if two variables are incremented at the same time

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  15. K

    I Is a Vector Field Equal to Zero if Its Contour Integral is Zero?

    I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?
  16. Math Amateur

    MHB The Elusive Field Structure of $\mathbb{R}^n$ Beyond $n=2$

    \mathbb{R}^n is a vector space but not a field because it lacks a suitable multiplication operation between pairs of its elements ... Why don't mathematicians define a multiplication operation between a pair of elements and investigate the resulting field ... For example ... why not define...
  17. CrosisBH

    I Why does this needle get an E field while this disk gets a D field?

    So I'm reviewing old lectures to prepare for an exam soon. This is about polarization. E fields, D Fields, etc. My professor labeled this diagram like so. The figures my professor drew are cavities in a dielectric if you can't read her handwriting. However, I can't seem to figure out why the...
  18. PainterGuy

    B Electrons in a nonmoving conductor and non-varying magnetic field

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  19. Z

    Evaluate the electric field for two slabs of charge

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  20. E

    Electric field in a spherical shell

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  21. V

    Increase in plasma energy from the increase in an ultraweak field

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  22. Z

    Find the electric field of a cylindrical charge

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  23. M

    Electron with circular trajectory in a magnetic field

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  24. garthenar

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  25. Vyrkk

    A Covariant derivative and connection of a covector field

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  26. M

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  27. karush

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  28. Diracobama2181

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  29. bryanso

    Why is the Electric Field outside this wire not zero?

    Not a homework. Just self-studying electromagnetism. I am stuck at understanding the very beginning of the solution steps in this example: The E as given in the solution is the field away from a long straight line with charge Lambda. That's clearly not the current configuration. E should...
  30. Moara

    Flux of Electric Field through a cone

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  31. S

    Effect of a charge's own field on itself (Feynman Lec. Vol. [I]-28 )

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  32. binbagsss

    A Hamiltonian background magnetic field, perturbed by electric field

    Hi I'm looking at Tong notes http://www.damtp.cam.ac.uk/user/tong/qhe/two.pdf deriving the Kubo Formula, section 2.2.3, page 54,I don't understand where the Hamiltonian comes from (eq 2.8). I tried a quick google but couldn't find anything. I'm not very familiar with EM Hamiltonians, any help/...
  33. T

    A Numerically Solving Scalar Propagation in Curved Spacetime

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  34. J

    I Does Moving a Wire Increase its Magnetic Field?

    Folks, I'm not sure if this is the correct forum topic for my question, but it seems to be close or related. Question: If you have a wire carrying current then it creates a magnetic field as in the diagram in the link below...
  35. Riotto

    A Canonical momentum ##\pi^\rho## of the electromagnetic field

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  36. B

    Not necessarily homework help but I do not understand Field lines

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  37. N

    Spin Hamiltonian of a hydrogen atom in a magnetic field

    qa
  38. M

    Work/Energy in a Magnetic Field

    Sorry if I am asking in the wrong fashion as I am new. The above questions are easily solvable: 1) U = -μBcos(0) 2) U = -μBcos(180) 3) W = ΔU = 2) - 1) My question is more related to some theory: where is this work/energy coming from since a magnetic force, to my knowledge can't do "work"...
  39. A

    Plasma current direction in the presence of an external field

    A simple question, I read that the particles in a plasma (protons , electrons) travel along the B field lines in a plasma which I know and they loop around the field lines each in opposite direction (clockwise, anticlockwise ) but both in the same direction. My question then is this. If I have...
  40. B

    Electric field of light emitted by atom, given the intensity

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  41. D

    Find the electric field inside and outside of a spherical shell superposition

    Hi! I need help with this problem. I tried to solve it by saying that it would be the same as the field of a the spherical shell alone plus the field of a point charge -q at A or B. For the field of the spherical shell I got ##E_1=\frac{q}{a\pi\epsilon_0 R^2}=\frac{\sigma}{\epsilon_0}## and for...
  42. F

    Engineering Determine the shunt field current in a magnetic circuit

    Problem Statement : Here's my attempt : * By assuming that the fringing and leakage effects are ignored. I find the flux density , the permeability and the reluctance of the iron , but then I get stuck . Any help would be greatly appreciated .
  43. jisbon

    Rotating cylinder rolling without slipping in a B field

    Firstly, I need to determine what the electric field is causing. Using left hand rule, the force due to the field is acting down the slope. Hence my FBD looks like: Where the two arrows pointing towards the right represent the force due to the field and weight of the cylinder. Since ...
  44. saadhusayn

    Expanding an action about a background field

    If we expand the action ##S[\phi]## about ##\varphi(x)##, ## S[\phi] = S[\varphi] + \int d^d x \frac{\delta S[\varphi]}{\delta \varphi (x)} f(x) + \frac{1}{2!}\int d^dx_1 d^d x_2 \frac{\delta^2 S[\varphi]}{\delta \varphi(x_1) \delta \varphi(x_2)} f(x_1) f(x_2) + \mathcal{O}(f^3)## But...
  45. S

    Correct statement about coils moving in and out of a magnetic field

    Option B is correct. Using Lenz's law, the direction of current flowing in coil 1 is counter clockwise while in coil 2 is clockwise. Option A is correct. Using Fleming's left hand rule, the resultant magnetic force acting on the both coils is to the left. I am not sure about option C and D. I...
  46. P

    A Einstein Tensor and Stress Energy Tensor of Scalar Field

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  47. S

    Correct statement about gravitational force, field and potential

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  48. M

    Field axioms - is there an axiom for multiplication with zero?

    Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom. I am trying to understand why the proof was not stopped at the encircled step. 1. Is there no axiom that says ## x \cdot 0 = 0 ## ? 2. Isn't the sixth...
  49. JorgeM

    Could an Electromagnetic field produce the "3-body" problem?

    .
  50. J

    A How can a spin moment align parallel to an external field?

    Hello everybody, Lets say we have an atom with an electron that have a spin. That spin can only pointing in 2 directions, in s_1=\hbar*1/2 and s_1=-\hbar*1/2 and therefore the magnetic moment of an electron is pointing on the opposite site. My Problem here is appearing when whe apply an...
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