Definition of a field in maths and physics Is the field as defined in analysis ( i.e a set with multiplication and addition defined on it such that they obey certain field axioms like closedness and associativity .... ) related by anyway to the " field" in physics ( e.g electric field, force field …).
Additionally, if they are not equivalent, you would do me great justice if you can refer me to some rigorous definition of "field" in physics. Thank You
 A: I don't  think that there is any connection; it is just a coincidence that a mathematician chose field for one concept and a physicist chose it for another. Such terms, especially in mathematics, are often completely arbitrary.  A mathematical field does not reasonable a place where you might play football or graze a horse, a ring in algebra does not look much like a thing that you might put on your finger, etc.
One way to see the arbitrary nature is to look at the corresponding terms in other languages.  A neat way to do this is to find the concept in English Wikipedia and then switch the language.  So, let's find the German for the two uses of field.
Here is the English article for field in the mathematical sense: Field (mathematics) and here is the German: Körper (Algebra). Now, let's find the day to day meaning of Körper (using Google Translate): Body.  So, again an arbitrary word but a quite different one.  This explains why $K$ is a popular symbol for a field in mathematics.
Let's do the same trick for the physics sense: Field (physics) and Feld (Physik).  This time, the meaning is close to the English.  So, a German speaker would be unlikely to ask a question like yours.
Playing the same trick for French gives a similar result: Corps (body) for the mathematical object and Champ (field) for the mathematical one.
Let's get more exotic.  Chinese for the mathematical object is 域 (Domain) and the physics object is 场 (field).
So, in my sample of four languages, I find that the physics term is quite consistent but the mathematics one varies.  Only in English (*) do they coincide.
(*) In my small sample, I expect that they match in some other languages.
I could not resist checking a few more languages.  Spanish and Portuguese follow French with body for the mathematical object but Italian follows English with field for both.  Danish follows German but with the less easily guessed word: Legeme.  I could not find any that did not use a close equivalent for the physics object.
A: In physics, a "scalar field" is essentially a function of position, or a number at every point. The temperature $T(x,y,z)$ at every point in a room is described by a scalar field.
A "vector field" is essentially a vector at every point.  It can have various units. So the gravitational field $\overrightarrow{g}$ has units of $m/s^2$ while the electric field $\overrightarrow E$ has units of $V/m$ and the magnetic field $\overrightarrow B$ is measured in teslas.
A gravitational field is in effect an arrow at every location in space pointing "down"--it is the definition of down, in fact.  The size of the vector indicates the strength of the gravitational field.
Mathematics is the map, and physics is the territory, in the case of physics fields. A scalar field or vector field is a mathematical object, one function or a set of functions with 3 inputs in three dimensional space. You can add these fields and so forth, do mathematical operations on them, but the physical phenomenon is the reality the model tries to describe.
As others have said, this is a very different use of the word "field" from how mathematicians use it to mean roughly "a set of things you can do arithmetic with" and more precisely follows various axioms of being a commutative group under addition with identity $0$ and nonzero elements being a commutative group under multiplication with identity $1$ and satisfying the distributive laws.
