Gapplet
A special automorphism of a finite field of order pa, where q = pa, a prime power, is the map x -> xp. We give the automorphism as a permutation of the nonzero elements g, g2, ..., gq-2, gq-1 = 1, of the field, where g is a primitive element.
Input a prime power q, e.g., 16. |
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Write out the permutation of the nonzero elements
corresponding to the automorphism x ->
xp.
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The permutation. Here the numbers refer to the exponents of the powers of g,
so (i,j,...) means g is mapped to
gj. Do you see why there are p-1 fixed points? | |
In fact, as we shall see later, the (cyclic) group (of order a) generated by this automorphism is the full automorphism group of the field.