# How many independent ways are to produce magnetic field?

I'm really confused with magnetic fields, After reading Purcell's book it seems that the magnetic field is due to choosing the wrong co-ordinate. Then I read this article which says no magnetic field is also a fundamental field. After a discussion on this post; I come up with three ways to produce a magnetic field that are

1. Moving charge produces a magnetic field
2. Magnetic field due to intrinsic magnetic moment of particles.
3. Changing the electric field produces a magnetic field.

I don't know if there are other ways. Now one more thing that confused me is changing the frames. If there is a field in one frame, it may or may not be in another. Two quantities that are invariant under Lorentz transformation are $$\mathbf{E}\cdot \mathbf{B}$$ and $$E^2-B^2$$. So from some, the things are clear through this post.

Now my question is How many ways are there through which a magnetic field can exist? By independent I mean here that if it exists in one frame then there is no frame where it doesn't. Please put all ideas that we know till now.