Is the addition and multiplication in the definition of field are universal or specific for each field

Question

In the definition of field Is the addition and multiplication in the definition of field are universal or specific for each field? I mean is there some definition of what are addition and multiplication ? are they work the same for every field like $\mathbb{C}, \mathbb{R}$ or their definition depend on the field itself ?

If they have a universal definition what is that definition ? How to rigorously define addition and multiplication all field ?

This question bothers me every time I use some general field $F$.

Answer 1

As long as the operations obey the restrictions given by the field axioms, the specifics are not otherwise defined.

To look at a slightly simpler situation, a group is a set with a single binary operation that obeys the axioms:

1. The operation is associative, i.e. $(ab)c = a(bc)$ for all $a, b, c$ in the group.

2. There exists an identity element $e$ such that $ae = ea = a$ for all $a$ in the group.

3. Every element has an inverse, i.e. for all $a$ in the group there is $a^{-1}$ such that $aa^{-1} = a^{-1}a = e$.

Groups include:

• The integers with addition.

• The positive real numbers with multiplication.

• Points on a unit circle with rotations.

• Arrangements of a Rubik's cube with combinations of moves.

In general, the idea is that we find (or define) a set with an operation, and then we prove that the operation obeys the group axioms. Having done that, we then know that any theorem that applies to groups applies to our set. Similarly, if we have a set with two operations that obey the field axioms, then we know all of the field theory applies to it even if the operations have no obvious connection to the addition and multiplication of real numbers.

Answer 2

Well I was also pondering upon the same question when I first got to know about fields and it has been long time so surely i have the answer.

Actually we don't need addition and multiplication in the definition of field but any two binary operation (operation which operate like a*b to give new element within the system or structure) which satisfy the very basic conditions like (F,*1) & (F\{0},*2) must be an abelian groups and these binary operations (*1 and *2) must be compatible i.e (F,*1,*2) must form a field.

But we usually take +, . in place of *1 and *2 because they are the most common and easy to explain with plenty of examples like (Q,+,.) or (R,+,.).

And in some place while defining fields we take them ( the + and .) just for sake of convenience as the associated binary operations.

the best examples are: Zp under modulo p addition and modulo p multiplication i.e (Zp,+p,.p) p being prime are finite fields .note that these addition and multiplication aren't same as normal addition and multiplication.