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. Haorong Wu

    Quantum Anyone tried "Problem Book in Quantum Field Theory" by Radovanovic?

    It is a wonderful book for learning QFT. Interesting problems with detailed solutions. I have tried the problems from chapter 1 to chapter 7. In most chapters, I could at least solve some part of the problems. But I got stuck in chapter 4, the Dirac equation. I could not solve any of the...
  2. willDavidson

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  3. willDavidson

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  4. PhysicsTest

    The net magnetic field at the center

    The current direction is as follows I think so much and do the right hand rule i get 0 at the center, but not sure why the answer is non zero. I have shown the directions of the magnetic fields, i have not shown the magnitudes of equal length but they all are equal. Why the answer is non zero...
  5. phywithAK

    How can I find conserved current for a Lagrangian involving vector fields?

    Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be : $$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$ I have seen examples based on the...
  6. P

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  7. LCSphysicist

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  8. J

    Modeling Uniform Field Gap Electrodes, Rogowski or others

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  9. PhysicsTest

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  10. M

    Find the electric field intensity from an infinite line charge

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  11. wcjy

    Electric field problem using Gauss' law: Point charge moving near a line charge

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  12. JD_PM

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  13. C

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  14. J

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  15. Z

    Electric Field Divergence of Monochromatic Plane Wave: Why is it Zero?

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  16. bln1230

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  17. peace

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  18. E

    B Understanding the active/passive transformation of a scalar field

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  19. J

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  20. AN630078

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  21. K

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  22. AN630078

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  23. archaic

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  24. madafo3435

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  27. J

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  28. cwill53

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  29. V

    I Object in or out of a circular field of view? (celestial coordinate system)

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  30. bryanso

    I Understanding Feynman's Relativistic Electric Field Equation

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  31. R

    Electric Field Between two Parallel Conducting Plates of Equal Charge

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  32. P

    Divergence of a radial field ##F=\hat{r}/r^{2+\varepsilon}##

    Following (1), \begin{align*} \text{div} F = \vec{\nabla} \cdot \vec{F} &= \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 F_{r}\right) \\ &= \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{1}{r^{2+\varepsilon}}\right) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}...
  33. P

    I Stability of circular orbits in an arbitrary central force field

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  34. agnimusayoti

    Electric field a distance z from the center of a spherical surface

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  35. R

    Using Gauss' Law to find the field at a point

    Attached is problem 23.03 from Halliday and Resnick. We have a sphere of uniform negative charge Q = -16e and radius R = 10cm. at the center of the sphere is a positively charged particle with charge q = +5e. We are supposed to use Gauss' law to find the magnitude of the electric field at...
  36. W

    I Renormalization of scalar field theory

    I was reading about the renormalization of ##\phi^4## theory and it was mentioned that in order to renormalize the 2-point function ##\Gamma^{(2)}(p)## we add the counterterm : \delta \mathcal{L}_1 = -\dfrac{gm^2}{32\pi \epsilon^2}\phi^2 to the Lagrangian, which should give rise to a...
  37. R

    Potential Energy of an Electric Dipole in a Uniform Field

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  38. DarkMattrHole

    B How does the electric field of an electron compare to its probability wave?

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  39. S

    How to calculate the induced charge in an electric field

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  40. Steven Ellet

    Medical How to find a scientist who researches a specific field?

    I have a Hypothesis on one potential cause for allergies and a potential cure based on that hypothesis. Unfortunately I have absolutely no way to test my hypothesis, Is there any way for me to get in contact with people studying that field?
  41. Tony Hau

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    The question is like this: The solution is like this: However, according to the equation for ##E_{dip}## , what I think is that it should be: $$E=\frac {1}{4 \pi \epsilon_o} \frac {qd}{d^3} \hat {\mathbf z} $$, where I take the centre of the sphere in figure 2 as the centre of the...
  42. A

    Is that field conservative? If yes, why is the work not null?

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  43. P

    What is the mistake in calculating the magnetic field in this problem?

    The problem is simple, but have one confusion, if i substitute the values given, I get ## B = \frac {10^{-7}(6*10^{-6})[(8*10^6 \vec j) \times (-0.5\vec j + 0.5 \vec k)]} {r^2} ## ## B = 48\mu T\vec i## First thing the answer does not match. I don't see the angle in calculations between ##\vec...
  44. Demystifier

    A Philosophy of quantum field theory

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  45. cwill53

    Electric Field and Continuous Charge Distribution

    I sort of understand the meaning of this integral, but I don't know how to evaluate it. I have never evaluated a volume integral. It would be very helpful if someone could explain in other words what this integral means and give an example evaluating it. This is from Purcell's Electricity and...
  46. aspodkfpo

    What happens when an uncharged object is brought into an electric field?

    Tecd
  47. preachingpirate24

    Electric Field inside the material of a hollow conducting sphere

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  48. P

    Find the magnitude and direction of the Magnetic field required

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  49. dRic2

    I Electric field inside a Superconductor

    I was reading chapter 3 of this book https://blackwells.co.uk/bookshop/product/Superconductivity-by-James-Arnett/9780198507567, which is a brief introduction to superconductivity. It is stated that inside a superconductor the Electric filed is always zero. This is deduced from the equation...
  50. Tony Hau

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    Today when I am reading Griffith's electrodymamics on surface charge and force on conductors, I have come across two very ambiguous terms: electric field at the surface and immediately outside the surface. The context of these two words is as follows: The electric field immediately outside is...
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