Abstract Algebra: Theory and Applications · Problem 22.8.2

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · finite-fields-sage

Problem

Corollary 22.1.11 says that the nonzero elements of a finite field are a cyclic group under multiplication. The generator used in Sage is also a generator of this multiplicative group. To see this, create a finite field of order 272^7. Create two lists of the elements of the field: first, use the .list() method, then use a list comprehension to generate the proper powers of the generator you specified when you created the field. The second list should be the whole field, but will be missing zero. Create the zero element of the field (perhaps by coercing 0 into the field) and .append() it to the list of powers. Apply the sorted() command to each list and then test the lists for equality.

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