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el3orian
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Why is the characteristic of a finite field a prime number?!
Hurkyl said:Short (but same) answer: char(F) is clearly not zero. If it were composite, then it's easy to find a nontrivial zero-divisor.
How can a field have a nontrivial zero-divisor?subGiambi said:Would you mind expanding on this explanation a bit? What is the significance of a nontrivial zero-divisor? Thanks!
The characteristic of a finite field is the smallest positive integer p such that adding p copies of any element in the field results in the zero element. In other words, it is the number of times one must add a given element to itself in order to get the additive identity (usually denoted as 0).
The characteristic of a finite field must be a prime number. This is because if the characteristic is not a prime number, then it can be factored into smaller integers, and this would violate the definition of characteristic as the smallest positive integer that results in the zero element when added to itself multiple times.
The prime characteristic of a finite field is important because it determines the structure and properties of the field. For example, the order of a finite field (the number of elements in the field) is pn where n is the dimension of the field over its prime subfield, and this is only possible if p is a prime number. Additionally, the prime characteristic also affects the behavior of the field's operations and polynomials defined over the field.
No, a finite field cannot have a characteristic of 0. This is because the characteristic of a field must be a positive integer, and 0 is not considered a positive integer. In fact, a field can only have a characteristic of 0 if it is infinite.
The characteristic of a finite field affects the structure of its subfields. For example, if the characteristic is a prime number p, then all subfields must also have a characteristic of p. This means that the subfields can only have elements that are multiples of p and cannot contain elements that are not divisible by p. This is important for understanding the structure of finite fields and their subfields.