Question about the definition of a field…

Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation?

Here's the definition we were taught in calculus for physicists class:

A field F is a non-empty set on which two binary operations are defined: an operation which we call addition, and denote by +, and an operation which we call multiplication and denote by $\cdot$ (or by nothing, as in a b = ab). The operations on elements of a field satisfy nine defining properties, which we list now...

And then of course you have the axioms of field.

• So... multiplication is nothing but repeated addition... So, in the real numbers, how do you add $\sqrt{3}$ to itself $\sqrt{2}$ times in order to compute $\sqrt{2}\times\sqrt{3}$? In the complex numbers, how do you add $i$ to itself $i$ times to get $-1$? In the field of rational functions with coefficients in $\mathbb{Q}$, how do we add $1+x$ to itself $x^3-2x+\frac{1}{2}$ times? – Arturo Magidin Mar 8 '12 at 17:10
• Sounds like the answer if it will be fleshed out. : ) – user21436 Mar 8 '12 at 17:11
• A previous question: If multiplication is not repeated addition... – Arturo Magidin Mar 8 '12 at 17:12
• In other words, multiplication is not "nothing more than repeating the same addition operation". Repeated addition is merely computational technique that works in a very specialized case. Except it's not a very good technique -- the reverse is far more useful: in a situation where you happen to be interested in repeated addition, multiplication is a useful computational technique. – user14972 Mar 8 '12 at 17:32

For an abstract field, $+$ and $\times$ are just symbols for two binary operations which need not be related in any way except by the distributive requirement(i.e. $a\times(b+c)=(a\times b)+(a \times c)$). We use $+$ and $\times$ because they represent operations in the fields we know and love best, the rational numbers, the real numbers and the complex numbers. You could use $\heartsuit$ and $\clubsuit$ , if you like them better.But, as Arturo pointed out, to think of multiplication as repeated addition in even these fields is dangerous. So, if your fields had elements which were say,sequences, it becomes worse, how do I add something like $(0,1,0,\cdots)$ to itself $(1,1,1\cdots)$ times?
But, this idea of "adding" elements $n$(for a natural integer) times has been thought about before and you might consider reading this to see how different things are in abstract fields. http://en.wikipedia.org/wiki/Characteristic_(algebra)
For example you can define a field where the elements are certain kinds of functions. Where the product of two functions $f$ and $g$ is the pointwise product $f \cdot g (x)=f(x)g(x)$. This product is in no way related to a sum of multiple copies of $f$.