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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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Define the trace by
Now define yet another mapping:
I know that this is a linear mapping. My syllabus stated the following I could't prove:
The first claim follows from the fact that the Galois group of Fpn/FpFpn/Fp is generated by the Frobenius map x↦xpx↦xp. Thus, from Galois theory, the subfield of FpnFpn fixed by G(Fpn/Fp)G(Fpn/Fp) is exactly FpFp. For your second statement, recall that Tr(yp)=Tr(y)Tr(yp)=Tr(y), now apply TrTr on both sides of the equation.
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