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[MadMojoMonkey]

Pardon my ignorance but I was the believe the Higgs Boson was proposed on the basis of field theory.

Yes. I see the confusion. Classical physics excludes Einstein's Relativity and Quantum Mechanics, which were developed in the early 1900's. Field theories were known for a while, as the Electromagnetic fields were well understood no later than 1873. As modern physics has developed, it has relied heavily on the use of fields. Fields are just a useful tool to keep track of things.

Rolling from memory, field theory is the most basic mathematic description of stuff. Fields are everywhere and have value where there is stuff, that stuff represents discrete jumps in specific fields. All fields interact with each other in different ways. The Higgs field exists and is everywhere nonzero, making it unique. Its interaction with the other fields yield mass.

A field exists everywhere in space, and is not discreet in this sense. There is no point in space where the field does not have a value. A discrete field would have a value at some points but not others. Like the integers are a discrete set of numbers on a continuous number line.

Some fields represent discrete quantities, but some represent continuous quantities.

Stuff is discrete in a sense. As in, "There's one electron in the ground state of that Hydrogen atom". But the field represents something like the position of the electron, which is not discrete. The properties of a particle are represented by probability functions. The position is smeared out over a volume, with high probability density regions and low probability density regions. Ultimately the position/momentum field for any particle is non-zero at all points in space.

Not all fields interact with each other. Electromagnetic fields do not interact with non-charged particles.

The explanation of mass is the unique property of the Higgs mechanism.

So, the other natural question is how indivisible are the Lepton, Boson and Quark?

That's another good question. I am quite curious myself. I'll get back to you on this.