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:
“The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice.”
—Blaise Pascal (1623–1662)
“I respect not his labors, his farm where everything has its price, who would carry the landscape, who would carry his God, to market, if he could get anything for him; who goes to market for his god as it is; on whose farm nothing grows free, whose fields bear no crops, whose meadows no flowers, whose trees no fruit, but dollars; who loves not the beauty of his fruits, whose fruits are not ripe for him till they are turned to dollars. Give me the poverty that enjoys true wealth.”
—Henry David Thoreau (1817–1862)