Examples of fields of characteristic 1 I'm a beginner in this topic, so this question seems stupid, but I think it's a doubt many beginners can have. 
I realized that I can't find examples of fields with characteristic 1, and as 1 is a prime number we can try to find examples of such fields.
I find a little bit strange this kind of field, it seems that it can have only the zero element, then it can't exist such fields
I said something wrong? 
Thanks
 A: The fields of characteristic $p$ are such that "$p=0$" by handwaving. Therefore, if $1=0$, the only field you can expect is the zero field, which is indeed, as you stated, a bit strange, for it is the only field with this property. For every other field, $1 \neq 0$. 
(EDIT : You can interpret my word "expect" in "the only field you can expect" this way : the definition of field that allows $1=0$ only adds the zero field to the possible fields, even though it is non-standard to do so, so we usually assume $1 \neq 0$ to get rid of this case. See the discussion in the comments for more details.)
Usually people do not consider $1$ as a prime, for it does not generate a prime ideal in the ring $\mathbb Z$. Now this again is a matter of definition ; we define the prime ideals those who satisfy some property and are not the whole ring. There are many other reasons why $1$ is usually not a prime, and you just found one of them. $1$ behaves significantly differently than the non-$1$ primes, so it is natural to leave it aside. 
Hope that helps,
A: According to Alain Connes, characteristic 1 only really shows itself clearly when you deal with semirings (only a commutative monoid under addition) and semifields (semirings in which non-zero elements form a group under multiplication). The only finite semifields are the finite fields and the 2-element Boolean algebra B, with "or" and "and" for + and *. (Note - this is not the 2-element field F_2, which has exclusive or for +.)
Although B does not satisfy 1=0, it does satisfy 1+1 = 1. Hence any B-algebra is a semilattice under addition.
