What is undefined times zero? Einstein's energy equation (after substituting the equation of relativistic momentum) takes this form:
$$E =   \frac{1}{{\sqrt {1 - {v^2}/{c^2}} }}{m_0}{c^2}
% $$
Now if you apply this form to a photon (I know this is controversial, in fact I would not do it, but I just want to understand the consequences), you get the following:
$$E = \frac{1}{0}0{c^2}% $$
On another note, I understand that after dividing by zero:


*

*If the numerator is any number other than zero, you get an "undefined" = no solution, because you are breaching mathematical rules.

*If the numerator is zero, you get an "indeterminate" number = any value.


Here it seems we would have an "indeterminate" [if (1/0) times 0 equals 0/0], although I would prefer to have an "undefined" (because I think that applying this form to a photon breaches physical/logical rules, so I would like the outcome to breach mathematical rules as well...) and to support this I have read that if a subexpression is undefined (which would be the case here with gamma = 1/0), the whole expression becomes undefined (is this right and if so does it apply here?).
So what is the answer in strict mathematical terms: undefined or indeterminate?
 A: I don't think you've quite come to terms yet with what undefined means. It means quite literally this expression has no meaning. So $1/0$ is literally meaningless even though it might look like it has meaning to our human brains, we haven't defined it, so it really has no meaning. For this reason, while we know that $x\cdot 0 = 0$ for any real number $x$, since $1/0$ is undefined, we don't even know that it is a real number, so we can't say $(1/0)\cdot 0 = 0$ nor can we say it is anything else for that matter. It is also undefined, because if we don't know what $1/0$ is, we can't know what anything made out of it is either.
Now of course one can try to extend definitions to define expressions that are currently undefined. However, for expressions such as the one you've given, the reason no one has done this is because there isn't a good way to do so. 
Now the phrase "indeterminate form" is honestly not a great phrase. Either something has a value or it doesn't. Usually it's used to describe limits that don't obviously exist and/or whose value isn't obvious. For example while
$$\lim_{x\to 0} \frac{\sin x}{x} = 1,$$
people often say that it has an indeterminate form, since $$\lim_{x\to 0} \sin x = 0 = \lim_{x\to 0} x.$$
Thus a naive approach to the limit would seem to give you that the limit is $0/0$, which is undefined. However, it turns out the limit is quite well-defined, it's just that the naive approach to taking limits of fractions doesn't work when the denominator goes to 0. 
Point being, your expression is undefined, and I would not describe it as indeterminate (though of course this depends on your definitions and where you're from). As I was taught, the phrase indeterminate form (if used at all) should be reserved for well-defined expressions such as limits whose existence or values are not obvious because a naive approach to taking such limits results in an undefined expression.
A: Undefined is not a number. There is no such number as undefined, for which you could define the multiplication operation.
You could extend a set of a numbers (the set of the real numbers or the set of the complex numbers) with a new element, what you call "undefined", and then define a multiplication on this set as usual. It might be possible (although there are major problems solve).
However, from this moment, you have a defined value what you still call "undefined". Well, paper can hold everything, but it does not really sound as useful mathematics.
