Question

Additive order of each element in field of characteristic zero

Answer 1

Answer:

If F is the field of p elements, F×F is a counterexample to all three.

Also, strictly speaking, it is never going to be possible to get all elements to have prime order. The zero element, at least, will have order 1.

Finally, the first sentence should include that the characteristic could be 0 (but if course, when it is nonzero and the ring is a domain, it is prime as you say.

Answer 2

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.

That is, char(R) is the smallest positive number n such that

if such a number n exists, and 0 otherwise.

The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive n such that

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity: mandatory vs. optional), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.

attempt,