This is actually exercise 3.7 from Computational Complexity: A Modern Approach.I need to prove that the NP-Completeness of 3-sat does not relativize, i.e. I need to show that that exists some oracle $A$ such that there exists $L\in \mathrm{NP}^A$ s.t. $L\not\leq_p\mathrm{3SAT}$, where the reduction can query the oracle for $A$.A lead of some kind would be most helpful (I know that using locality in the proof for Cook-Levin Theorem ruins the relativization, but it doesn't help me find an example).Another question popped on the way (I hope it's ok): why is $\mathrm{EXP}$ not low on itself, i.e. why$\mathrm{EXP^{EXP}}\neq \mathrm{EXP}$? Cant I simply simulate the machine $M_L$ deciding some $\mathrm{EXP}$-Complete oracle $L$, for exponential number of steps and replace querying the oracle for input x with the output $M_L(x)$. Suppose $M\in \mathrm{EXP^L}$ runs in $2^{n^c}$ time, and $M_L$ runs in $2^{n^d}$, the machine replacing the oracle queries runs in at most $2^{n^c}\cdot2^{n^d}$ and is therefore in $\mathrm{EXP}$.Thanks! 
Relativization of NP-completeness
complexity theory;np complete;oracle machines;relativization