Unformatted text preview: • • Staple ! Sets and Logic MHF32022787 Project-X Prof. JLF King 11Jan2012 OYOP: Your 2 essay(s) must be typeset , and double or triple spaced. Use the Print/Revise cycle to produce good, well thought out, essays. Start each essay on a new sheet of paper. Do not restate the problem; just solve it. Due: By noon , on Friday, 09Dec2011. Please write DNE in a blank if the described object does not exist or if the indicated operation cannot be performed. X1: Your goal is to prove: Eighth-root thm: For each oddprime p , the con- gruence x 8 ≡ p 16 admits a solution. † : In your WU, use ∼ for ≡ 4 and ≈ for ≡ 8 , if needed. But use ≡ p or ≡ for congr-mod- p . i FTSOC, suppose you have a p with no solution to x 8 ≡ p 16. Prove that 2 ∈ NQR p and 1 ∈ QR p . Use LSThm to compute hh p ii 8 as a non-neg residue. ii Let r be a p-sqroot of 1. Use LST to prove that r ∈ QR p . But use a different part of LST to prove that r ∈ NQR p . Contradiction , QED....
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