Let F be a field. The intersection of all subfields of F is itself a subfield, called the prime subfield of F.

Proposition The prime subfield of F is either isomorphic to Q or Zp, for a prime p.

Definition We say that F has characteristic zero if it has prime subfield isomorphic to Q and characteristic p if it has prime subfield isomorphic to Zp.

Proof of the proposition: Let P be the prime subfield. Since it is a subfield of F it contains 1F. We need some notation. If n is a positive integer define n.1F=1F+....+1F for n copies of 1F added together. We define -n.1F=-(n.1F) and 0.1F=0F. With this notation it is easy to check that
n.1F+m.1F=(n+m).1F
(n.1F)(m.1F)=(nm).1F

There is a function f:Z-->P defined by f(n)=n.1F. The two previous formulae show that f is a ring homomorphism. The kernel is an ideal of Z. Since Z is a principal ideal domain there exists a (nonnegative) integer n such that the kernel consists of all multiples of n.

There are two possibilities. Either

  1. n>0
  2. n=0
Consider case 1. first. Note that we cannot have n=1 since we cannot have 1F=0F. Now I claim that n must be prime. For if n=rs, with r,s> 1, then we can think about
(r.1F)(s.1F)=n.1F=0
By the choice of n we have that r.1F and s.1F are nonzero. Since P is a field r.1F is therefore a unit. Multiplying the equation by its inverse we deduce that s.1F=0. This contradiction shows that n=p is prime.

Thus by the first isomorphism theorem the image of f is isomorphic to Zp. The image of f is therefore a field. Since it is a subring of F contained in P it must coincide with P, proving one part of the proposition.

Consider case 2. Since the kernel is {0} f is injective. Consider the field of fractions of the image of f as a subring of F. Clearly this coincides with P and is isomorphic to Q.

Examples

  • Q,R,C all have characteristic zero, as does any subfield of C.
  • A finite field has characteristic p, for a prime p (there isn't room in a finite field for Q). But not all fields of finite characteristic are finite. For example, consider rational functions in one variable K(x) over a finite field. This field has finite characteristic, but infinitely many elements. Another example is the algebraic closure of a finite field.
A slightly sloppier, although more intuitive notion of characteristic, involves iterating a ring R's operators using its two identity elements.

We start off with a ring's additive identity 0R. We then add the ring's multiplicative identity 1R to this value. We iterate the operation over and over, always adding 1R to the previous result.

If we ever reach the ring's additive identity 0R again within a finite number of steps (call the number n), we say that the ring has characteristic n. Otherwise, the ring has characteristic 0.

Thus, the following field stolen from artermis enteri's writeup on field:

+ | 0 1     * | 0 1
--+----     --+----
0 | 0 1     0 | 0 0
1 | 1 0     1 | 0 1

has characteristic 2.

I have a personal vendetta against the definition of 'characteristic' as stated above. First I'll state my (proposed) alternate definition, then I'll give some reasons for my dislike.

The Characteristic of a ring with identity

Given a ring R with identity e, let n be the additive order of e. By definition of additive order, n is a positive integer, or positive infinity. Let this n be known as the characteristic of R. (Part of the reason for this definition is that, for a given rR, n*r = n*r*e = r*(n*e) = 0R, so n is (a multiple of) the additive order of each element of R.)

My reasons for this definition include the following:
* There are no rings with identity which can also have characteristic 1. If we allow a ring to have characteristic 0 we have created a gap in the set of possible characteristics.
* 0 is not a positive integer. This also conflicts with how we define characteristic. (On the other hand, I admit that infinity is not a positive integer... at least it's positive though.)
* Using infinity for a characteristic matches up more closely with the usual idea behind the order of an element.

These are my ideas and reasons. Do not confuse them with those of the "professionals". Also, please glance at my Odd-Even Theorem writeup for more information regarding why I choose this definition of characteristic. As an alternative (if my ideas cause too much grief/stir), I might decide to use 'focus' as my term for the number described above. It makes no exceptional difference to me.


In unrelated news, we have Moore's theorem on integral domains:

"Let R be an integral domain and n be its characteristic (as defined in this writeup). Then for all positive integers t, either n divides t or for all a, b in R, t*a=t*b implies a=b."

Proof

Let R be an integral domain and n be its characteristic (as defined in this writeup). Let t be a positive integer. If n divides t then we are done, except we note that in this case, there exist a and b in R such that t*a=t*b(=0), but a != b ("!=" is "does not equal"). Now we consider the case where n does not divide t. Suppose that there exist a and b in R such that t*a=t*b but a != b. Then t*a-t*b=0R=t*(a-b), which means that t is a potential candidate for the characteristic of R. Let c=a-b. n does not divide t, so that means that d=gcd(n,t)<n. From number theory, we know that there exist integers x and y such that n*x+t*y=d. We already know that t*c=t*(a-b)=0R. We further note that n*1R*c=n*c=0R. Now we note that d*c=(n*x+t*y)*c=0R+0R=0R, so that d is the real characteristic of R, and d divides t. But we already said that n was the characteristic of R, so this is a contradiction (since d < n and the characteristic is a 'smallest positive integer such that...'). Therefore (in this case) t*a=t*b implies a=b. Note that characteristic infinity presents a minor problem in this case. So suppose that there exist a and b such that t*a=t*b but a != b. Then, as before, we have t*(a-b)=t*c=0R. t is less than infinity, and we again have a contradiction. So such an a and b cannot exist. QED

Char`ac*ter*is"tic (?), a. [Gr. : cf. F. charact'eristique.]

Pertaining to, or serving to constitute, the character; showing the character, or distinctive qualities or traits, of a person or thing; peculiar; distinctive.

Characteristic clearness of temper. Macaulay.

 

© Webster 1913.


Char`ac*ter*is"tic, n.

1.

A distinguishing trait, quality, or property; an element of character; that which characterized.

Pope.

The characteristics of a true critic. Johnson.

2. Math.

The integral part (whether positive or negative) of a logarithm.

 

© Webster 1913.

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