Solution Found!
Suppose that F is a field of order 1024 and List the
Chapter 22, Problem 34E(choose chapter or problem)
QUESTION:
Suppose that \(F\) is a field of order 1024 and \(F\)* = \(\langle\alpha\rangle\). List the elements of each subfield of \(F\).
Questions & Answers
QUESTION:
Suppose that \(F\) is a field of order 1024 and \(F\)* = \(\langle\alpha\rangle\). List the elements of each subfield of \(F\).
ANSWER:Step 1 of 2
We need to list the elements of the subfield of F if F is a field of order 1024.
Notice that \(F=2^{10}\)
We know a theorem that states that for each divisor m of n, GF(pn) has a unique subfield of order (pm). Moreover these are only subfields of GF(pn).
Then the four subfields are:
\(\left\{G F(2), G F\left(2^{2}\right), G F\left(2^{5}\right), G F\left(2^{10}\right)\right\}\)