Field trace

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.

Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,

is a K-linear transformation of this vector space into itself. The trace, Tr_L/K(α), is defined as the (linear algebra) trace of this linear transformation.

For α in L, let σ₁(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then

If L/K is separable then each root appears only once and the coefficient above is one.

More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.

where Gal(L/K) denotes the Galois group of L/K.

Let ${\displaystyle L=\mathbb {Q} ({\sqrt {d}})}$ be a quadratic extension of ${\displaystyle \mathbb {Q} }$. Then a basis of ${\displaystyle L/\mathbb {Q} {\text{ is }}\{1,{\sqrt {d}}\}.}$ If ${\displaystyle \alpha =a+b{\sqrt {d}}}$ then the matrix of ${\displaystyle m_{\alpha }}$ is:

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