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Number Theory Test 2
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Number Theory Test 2
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25 Questions

1. Let alpha = a+mZ be a unit and gamma be a primitive root such that alpha = gamma^t for some non-negative integer t. Then alpha is a square if and only if...

2. For integers with gcd(m,n) = 1, phi(mn) =

3. Fermat's little theorem

4. Let m1, m2, ... , mr be pairwise relatively prime. Then a congruent to b (mod m1m2...mr) if and only if

5. The congruence aX congruent to 1 (mod m) has a solution if and only if...

6. If a is congruent to b (mod p) then (a/p)...

7. What is the order of a unit alpha = a+mZ where gcd(a,m) = 1? (in terms of modulo)

8. If every non-zero element in Z/mZ has a multiplicative inverse then Z/mZ is a ________

9. A congruence of integers is equivalent to an equality of congruence classes (T/F)

10. If p is an odd prime then the number of quadratic residues in Z/pZ is...

11. If p is an odd prime and p !| a, then the polynomial X^2 - a has either....

12. If gamma is a primitive root modulo the odd prime p, then... gamma^(p-1)/2 =

13. Let alpha = a+mZ be a unit and t a non-negative integer. Alpha^t = 1 if and only if....

14. How do the rings Z/mZ and Z differ? (3)

15. Z/mZ is not a ring (T/F)

16. Wilson's Theorem

17. What is the Legendre Symbol (a/p)?

18. a is congruent to b (mod m) if and only if...

19. Let p be prime. What is another way of writing -1+pZ?

20. Z/pZ where p is a prime is a ring but NOT a field (T/F)

21. Let m be a natural number. Then phi(m) =

22. Let alpha = a+mZ be a unit. For non-negative integers s and t, alpha^s = alpha^t if and only if...

23. What is a quadratic residue or square modulo p?

24. Let f(X) be in Z[X]. We define f(alpha = a+mZ)

25. Let alpha = a+mZ be a unit and t be a non-negative integer.

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