Computes the field trace \(\mathrm{Tr}_{L / K}(x)\) of the elements of \(x\).

Returns¶

The field trace of \(x\) in the prime subfield \(\mathrm{GF}(p)\).

Notes¶

The self array \(x\) is over the extension field \(L = \mathrm{GF}(p^m)\). The field trace of \(x\) is over the subfield \(K = \mathrm{GF}(p)\). In other words, \(\mathrm{Tr}_{L / K}(x) : L \rightarrow K\).

For finite fields, since \(L\) is a Galois extension of \(K\), the field trace of \(x\) is defined as a sum of the Galois conjugates of \(x\).

References¶

• https://en.wikipedia.org/wiki/Field_trace

Examples¶

Compute the field trace of the elements of \(\mathrm{GF}(3^2)\).

In [1]: GF = galois.GF(3**2, display="poly")

In [2]: x = GF.elements; x
Out[2]: 
GF([     0,      1,      2,      α,  α + 1,  α + 2,     2α, 2α + 1,
    2α + 2], order=3^2)

In [3]: y = x.field_trace(); y
Out[3]: GF([0, 2, 1, 1, 0, 2, 2, 1, 0], order=3)