Download Math 110 Midterm Examination

Math 110 Midterm Examination
Due at the beginning of class on Friday, February 16. This is an open book
exam, but you should not consult with other students.
1. Show that the quadratic formula
x=
−b ±
√
b2 − 4ac
2a
holds for solutions of the quadratic equation aX 2 + bX + c = 0 in Z/p,
where p is an odd prime, in the sense that if b2 − 4ac has two square roots,
the solutions are given by the above formula, if b2 − 4ac = 0 then there is
only one solution, and otherwise, there are no solutions to the equation.
2. Show that in Z/p, the equation x3 − 1 = 0 has at most 3 solutions.
3. How many solutions does the equation x5 = 1 have mod 341? Explain
your reasoning.
4. Construct addition and multiplication tables for a field with 9 elements.
5. Show that there is no prime p of the form 3k + 1, for which there is a
primitive root α so that the equations
α3 = 175
α7 = 245
both hold mod p.
6. Suppose that in Z/n, we have x3 = y 3 = z 3 = w3 , and the elements
x, y, z, w are pairwise distinct. Show that n is composite.
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