Frobenius For Finite Fields
Let Fq be the finite field of q elements, where q=pe. F fixes Fp by the argument above. If e=2, then F2, the second iterate of Frobenius, fixes p2 elements, so it will fix Fp2. In general, Fe fixes Fpe. Furthermore, F will generate the Galois group of any extension of finite fields.
Read more about this topic: Frobenius Endomorphism
Famous quotes containing the words finite and/or fields:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“On fields all drenched with blood he made his record in war, abstained from lawless violence when left on the plantation, and received his freedom in peace with moderation. But he holds in this Republic the position of an alien race among a people impatient of a rival. And in the eyes of some it seems that no valor redeems him, no social advancement nor individual development wipes off the ban which clings to him.”
—Frances Ellen Watkins Harper (1825–1911)