Trace ( eng. Trace ) - displaying the elements of a finite field extension in the original field K , defined as follows:
Let E be a finite extension of degree K , Is an element of the field E. Since E is a vector space over the field K , this element defines a linear transformation . A matrix can be associated with this transformation in some basis. The trace of this matrix is called the trace of the element α . Since in a different basis a similar matrix with the same trace will correspond to this mapping, the trace does not depend on the choice of basis, that is, each trace element is uniquely associated with its trace. It is designated or, if it’s clear what kind of extension it’s .
- 1 Trace Properties
- 2 Expression of a trace in terms of automorphisms E over K
- 3 Example
- 4 See also
- 5 Literature
- If E is a separable extension , then - nonzero functional, if inseparable, then .
- The trace is transitive, i.e. for the extension chain we have
- If Is a simple algebraic extension and Is the minimal polynomial α, then
Expression of a trace through automorphisms E over K
Let σ 1 , σ 2 ... σ m be all automorphisms of E leaving the elements of K fixed. If E is separable, then m is equal to the degree of [E: K] = n . Then for the trace there is the following expression:
If E is inseparable then m ≠ n , but n is a multiple of m , and the quotient is a power of characteristic p: n = p i m .
Let K be the field of real numbers , and E the field of complex numbers . Then the trace of the number is equal to . The trace of a complex number can be calculated by the formula , and this is in good agreement with the fact that complex conjugation is the only automorphism of the field of complex numbers.
- Norm (field theory)
- Van der Waerden B. L. Algebra-M :, Science, 1975
- Zarissky O. , Samuel P. Commutative Algebra Vol. 1-M :, IL, 1963
- Leng S. Algebra-M :, World, 1967