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. Hamiltonian

    Calculation of the field due to a dipole at an arbitrary point

    I know how to derive field using ##E = -\nabla V## in polar coordinates and doing so gave me $$E = (kP/r^3)(1 + 3cos^3\theta)^{1/2}$$ now I am trying to derive ##E## at point P using the fields produced by +ve and -ve charge respectively and taking components of each along the radial direction...
  2. P

    Engineering Challenging situation in finding a job in my field

    Hello, I would like to present my situation and maybe hear some suggestions on how to overcome the challenge I am facing: I have a PhD in a STEM field from Canada (I am an immigrant), and after I received my PhD, I worked 18 months in the academia as a postdoctoral researcher. During my...
  3. F

    A Quintessence field equation of state

    Why the equation of state parameter w for a quintessence field with potential m²Phi² is not in between 0 and 1/3 depending on the energy of its particles relative to the mass of the field instead of the w resulting from formula 9 of https://arxiv.org/pdf/1504.04037.pdf which seems to be...
  4. Athenian

    Finding the Magnetic Field (B_0) on a Graph [NMR]

    Background Information (Not Strictly Necessary): As a quick recap, the graph I am dealing with is a semi-logarithmic graph of free induction decay (FID) amplitude as a function of time. To acquire the value for ##{T_2}^*## (and its uncertainty) in the graph, I used the below equation to do so...
  5. R

    Vector Field associated with Stereographic Projection

    I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...
  6. S

    Electric Field Intensity: Intersecting E-fields Explained

    This the photo of my textbook, Here In the derivation of Electric Field Intensity due to an infinitely long straight uniformly charged wire - In the figure 1(c).20 Vector E1 and vector E2 are electric fields at point P due to the two line elements as shown in the figure 1(c).20. But, in this...
  7. M

    B Spin collapse in a magnetic field

    Basic descriptions of spin such as the beginning of Lindley's "Where does the weirdness go" state that an electron's spin doesn't exist or is "indeterminant" until measured (e.g. passed through a Stern-Gerlach field). However, isn't the magnetic field nonzero essentially everywhere (albeit...
  8. Mayan Fung

    Electric field acting on the source charge

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  9. JD_PM

    Varying an action with respect to a scalar field

    Let us work with ##(-+++)## signature Where the metric ##g_{\mu \nu}## is the flat version (i.e. ##K=0##) of the Robertson–Walker metric (I personally liked how Weinberg derived it in his Cosmology book, section 1.1) \begin{equation*} (ds)^2 = -(dt)^2 + a^2(t) (d \vec x)^2 \end{equation*}...
  10. Mayhem

    Electrostatics: Coloumb's law and the Electric Field

    Hi. How does the charge ##q## relate to the charges ##q_1, q_2## (see equations). For example, if ##q_1## is an electron and ##q_2## is a proton, is ##q## just a product or sum or something else of the two? Also, in Columb's law, how would I conceptualize charges ##q_1, q_2## in a system of a...
  11. D

    Induced Electrical Field (Maxwell-Faraday's Law)

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  12. JD_PM

    A Renormalizability conditions for a real scalar field in d dimensions

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  13. P

    Work Done on A Proton in an Electric Field

    W=-qEd =-(1.6*10^-19)(23)(0.75) = -2.76*10^-18 J However, the answer is 2.76*10^-18 J. Why is the word done positive and not negative? Since it's traveling in the same direction as the electric field, shouldn't it be negative work?
  14. H

    Motion of a particle in a magnetic field

    Hi, I have to find the motion of a particles ##(x,y,z)##. However, I'm not sure where to begin. Is it correct to split the problem and first find what's the motion in the x direction then y and z. For exemple, ##m \frac{d^2x}{dt^2} = -kv_{0x} + qv_{0x}B sin 90 ## ##m\int\int...
  15. K

    I Does the external field effect prove MOND over dark matter?

    this paper, Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies Kyu-Hyun Chae1 , Federico Lelli2, Harry Desmond3, Stacy S. McGaugh4, Pengfei Li4, and James M. Schombert5 Published 2020 November 20 claims a 4 sigma...
  16. H

    Determining electric field using gauss's law--different distributions

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  17. A

    ِAngular distribution of a scattered field

    SA I study the reflection from a rough surface and get the field as the attached figure my question can I use the field distribution to get the angular distribution of the scattered field and how can I project the field to plane waves?
  18. A

    Exploring the Inconsistencies of Work and Energy on an Infinitely Thick Sheet

    My doubt is in the mathematics part. As the force will be contrary to the direction of the field due to the negative charge, the particle will be attracted to the origin, so I assumed that the speed will increase (this is also found in the exercise response). I then applied an integral for the...
  19. naviakam

    I Estimating Particle Number in an Electric Field

    In an electric field ##E## if the energy of charged particle (ion) is formulated as ##W=qEr## where ##r## is the distance, how the number of particles is estimated?
  20. Fikremariam

    Electric Field -- Potential between the ends of a 2 meter stick

    I tried to find the charge from the formula v=k q/r and apply it to find the field but couldn't be sure
  21. robphy

    Insights The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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  22. P

    Work done by electric field to move capacitor plate

    so the result I got is the energy lost by the field right??
  23. P

    Problem on induced electric field, electrodynamics

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  24. T

    Electric field inside a charged ring

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  25. A

    I Electric Field seen by an observer in motion

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  26. nmsurobert

    High School Magnetic Field Lab

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  27. CPW

    Magnetic field does no work -- true in all branches of physics?

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  28. J

    I Does the Dilaton field change particle mass?

    The low-energy effective action of the bosonic string is given by: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big)$$ where ##H_{\mu\nu\lambda}=\partial_\mu B_{\nu\lambda}+\partial_\nu...
  29. R

    I The Central Field Approximation for Many-Electron Atoms

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  30. T

    What is a Solenoidal Vector Field?

  31. T

    POTENTIAL DIFFERENCE AND ELECTRIC FIELD HOMEWORK

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  32. A

    Quantum Resources for learning Quantum Field Theory

    hello :) i would very much like study some quantum field theorie, but have not previously study any regular quantum mechanic (i am not so interest in regular quantum mechanic, but more the relativistic theories). so i ask, this is possible or not? to what extent knowledge of regular quantum...
  33. e0ne199

    Engineering Need help about the magnetic field near an infinite current sheet

    here is the question, don't mind about point (a) and (b) because i have solved them already...the main problem is the question on point (c) : so far, what i have done is : H = 2.7*0.1-(1.4*0.15+1.3*0.25) = -0.265 az A/m which is the wrong answer compared to the solution provided from the...
  34. H

    The magnitude of the magnetic field at the center of the loops is....

    I calculate like this and I think the answer is E not D. N( µoI/2a + µo2I/2*2a) = 2µoNI/2a = µoNI/a
  35. M

    I How to determine matching coefficient in Effective Field Theory?

    Assume that I have the Lagrangian $$\mathcal{L}_{UV} =\frac{1}{2}\left[\left(\partial_{\mu} \phi\right)^{2}-m_{L}^{2} \phi^{2}+\left(\partial_{\mu} H\right)^{2}-M^{2} H^{2}\right] -\frac{\lambda_{0}}{4 !} \phi^{4}-\frac{\lambda_{2}}{4} \phi^{2} H^{2},$$ where ##\phi## is a light scalar field...
  36. H

    Bending proton beams under magnetic field

    I haven't taken a physics courses in some time and I'm having trouble getting started with this textbook question. I know that there will be relativistic effects present, but I can deal with that. The problem is how I can approach the problem. I initially thought of a geometric way to set up...
  37. M

    LaTeX Delete \entry Field in Resume Template: CurVe cv class

    Referencing this resume template, do you know how to delete the \entry field (dates) within the rubric (so that the text will align left without indentation)? I only want this for one rubric, but am unsure how to do this. Thanks so much!
  38. AN630078

    Drawing Radial Field Equipotentials and Field Lines

    I have just attached a standard depiction of a radial field as one may similarly choose to draw it. So I understand that the gravitational field strength in a field is defined as the force per unit mass at that point. The field lines in a radial field move further apart further away from the...
  39. Ibby321

    Vertical motion through an electric field

    I know that the answer is 0 J (no NET work was done) because there is symmetry to the problem and this symmetry comes from the fact that the direction of force changes, BUT I don’t know why the force changes (I have an idea; TBD below in #4). When I did this problem I thought I could find the...
  40. Jarvis323

    Magnetic field excursion about 41,000-42,000 years ago

    A global environmental crisis 42,000 years ago https://science.sciencemag.org/content/371/6531/811
  41. rcgldr

    Factoring polynomials with finite field coeffcients

    I'm not sure if I should post this here or in the mathematics section. I'm trying to find a way to implement a mapping of a larger finite field such as GF(2^64) to a composite field GF((2^32)^2). Let f(x) be a primitive polynomial for GF(2^64), with 1 bit coefficients. If the coefficients of...
  42. I

    Electric Field from a Charged Semicircular Rod

    Charge QQ is uniformly distributed along a thin, flexible rod of length LL. The rod is then bent into the semicircle shown in the figure (Figure 1).Find an expression for the electric field E⃗ E→ at the center of the semicircle. Hint: A small piece of arc length ΔsΔs spans a small angle...
  43. Costweist

    Changes in electric field lines as a result of an oscillating charge

    The last couple of days I’ve been troubled with a specific part of electromagnetism. How will electric field lines be affected by an oscillating charge? More specific, what will happen with the “amplitude” of a wave in an electrical field line as the wave propagate away from the charge? 1. Will...
  44. Y

    Conservation of Energy on Current-Carrying Wire in Magnetic Field

    So force on a current carrying wire = ILxB. If I have a bunch of bar magnets making a uniform magnetic field of strength B, then a 1 meter long wire of 0 ohms carrying 1 Amp, the force on that wire is (1)(1)xB = 1B. If I let that force move the wire for a time T, let's assume the wire moved a...
  45. K

    B How the Earth's Magnetic field deflects the solar wind

    On a previous thread (now locked) I was wondering about how, precisely, the Earth's magnetic field protects us from the solar wind. Posting this here because what I wrote in that thread is very wrong, and I think it's an interesting topic. I had a hell of a time finding good information. I...
  46. A

    Electromagnetism - movement of a coil in a magnetic field

    So my idea was that to reach the equilibrium position, the final moment of force has to be 0 (so in the end the forces will “eliminate” each other). And I found the equation Fm=B*I*l*sinα, which should characterize the force, which affects wire with the current in a magnetic field, and Fleming’s...
  47. B

    How to create an Electric Field?

    Dear friends, First of all I have one question! As per Figure 1, how to implement electrical connection in real life which are seen inside Red Box? and what is the meaning of grounding the other terminal? Figure 1 And the second thing is that, I want to create and electric field on copper...
  48. B

    Do neuron electrical signals generate an electromagnetic field or wave?

    I'm not sure where this belongs, I'm guessing biomedical, but I'm interested from a physics perspective. Do neurons generate an electromagnetic field? In other words, all the neural activity in the brain, does it generate electromagnetic fields? If so, what are the details of these fields? I...
  49. A

    Charges, rod and magnetic field

    I have some difficulties in solving this problem. This is what I did. I wrote down the equation of motion for the masses. For the first point \begin{equation} m\ddot{\textbf{r}}_1=\textbf{F}_1=q\dot{\bar{\textbf{r}}}_1\times...
  50. H

    Question about the force exerted by a uniform electric field on a dipole

    I think the answe should be C as shown in the attachment.
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