Absolute value (algebra)

In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. If D is an integral domain, an absolute value is a mapping | | from D to the real numbers R satisfying

$|x| \geq 0$
$|x| = 0\Leftrightarrow x = 0$
$|xy| = |x||y|$
$|x+y| \leq |x|+|y|$

Types of absolute value

If |x+y| satisfies the stronger property

|x+y| \leq \max(|x|; |y|)

then | | is called an ultrametric or non-Archimedean absolute value; otherwise the absolute value is Archimedean. If |x|=1 for all nonzero values, the absolute value is trivial, otherwise it is nontrivial.

Places

If | |₁ and | |₂ are two absolute values on the same integral domain D, then the two absolute values are equivalent if ₁ < 1 if and only if

Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p. For a given prime p, the p-adic absolute value of the rational number $q = p^n \frac{a}{b}$ , where a and b are integers not divisible by p, is $|p^n \frac{a}{b}|_p = p^{-n}$ . Since the ordinary absolute value and the p-adic absolute values are normalized, these define places.

Valuations

If for some ultrametric absolute value we define $\nu (x) = - \log_b(|x|)$ for any base b>1, and add to this the special value ν(0)=∞, which is ordered to be greater than all real numbers, we obtain a function from D to R ∪ ∞, with the following properties:

$\nu(x) \leq \infty$
$\nu(x) = \infty \Leftrightarrow x=0$
$\nu(xy) = \nu(x) + \nu(y)$
$\nu(x+y) \geq \min(\nu(x), \nu(y))$

Such a function is known as a valuation. Some authors call it an exponential valuation, and call what Wikipedia calls an absolute value a valuation; Wikipedia is following the terminology of Bourbaki.

Completions

Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 there is a positive integer N such that for all integers m, n > N one has $|x_m-x_n| < r$ . It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a ring. One can also define null sequences as sequences of elements of D such that |a_n| converges to zero. Null sequences are a prime ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain. The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value | |.

Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we can either show that null sequences form a maximal ideal, or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element.

Another theorem of Alexander Ostrowski has it that any field complete with respect to the usual Archimedean absolute value is isomorphic to either the real or the complex numbers. Ultrametric complete fields are far more numerous, however.

Fields and integral domains

If D is an integral domain with absolute value | |, then we may extend the definition of the absolute value to the field of fractions of D by setting

|x/y| = |x|/|y|

On the other hand, if F is a field with ultrametric absolute value | |, then the set of elements of F such that |x| ≤ 1 defines a valuation ring, which is a subring D of F such that for every nonzero element x of F, at least one of x or x^-1 belongs to D. Since F is a field, D has no zero divisors and is an integral domain. It has a unique maximal ideal consisting of all x such that |x|<1, and is therefore a local ring.

References

Nicolas Bourbaki, Commutative Algebra, Addison-Wesley, 1972

Gerald J. Janusz, Algebraic Number Fields, second edition, American Mathematical Society, 1995

Abstract algebra