Download Raji 5.2, Primitive roots for primes: 8. Let r be a primitive root of p

Raji 5.2, Primitive roots for primes:
8. Let r be a primitive root of p with p
a primitive root.
1 (mod 4). Show that
r is also
I suppose p is a prime. Indeed, 2 is a primitive root modulo 9, but
not.
2 is
Write p = 4m + 1. As r is a primitive root, the numbers r; r2 ; r3 ; : : : ; r4m
are a complete set of nonzero residues modulo p. Note that r2m 6= 1, but
2
r2m = 1, so r2m = 1 because p is a prime.
Let s = r and consider the number s; s2 ; s3 ; : : : ; s4m . Note that if i is
even, then si = ri while if i is odd, then si = ri . Suppose si = 1 for
some i such that 1 i 4m. If i is even, then ri = 1 so i = 4m. If i is
odd, then ri = 1. But r2m = 1, so i cannot be odd. So the only i with
1 i 4m such that si = 1 is i = 4m, which means that s is a primitive
root.
9. Show that if p is a prime and p
such that x2
1 (mod p).
1 (mod 4), then there is an integer x
Didn’t we prove that in 8? The number x is rm .
Raji 5.3, The existence of primitive roots:
1. Which of the integers 4, 12, 28, 36, 125 have primitive roots?
Which are 2, 4, pe , or 2pe for an odd prime p? Only 4 and 125, right?
2. Find primitive roots of 4, 25, 18.
For 4, the primitive root is 3. For 25, I would …rst try 2. The powers of
2 are 2; 4; 8; 16; 7; 14; 3; 6; 12; 24 = 1, so 210
1 and ord25 2 = 20 =
' (25). For 18, the …rst candidate is 5, whose powers are 5; 7; 1; 5; 7; 1.
As ' (18) = ' (2) ' (9) = 6, we see that 5 is a primitive root of 18.
3. Find all primitive roots modulo 25.
We know that 2 is a primitive root. The others are 2i where i is relatively
prime to ' (25) = 20. So the primitive roots are 2, 23 , 27 , 29 , 211 , 213 , 217 ,
and 219 . Eight of them? Sure, because ' (20) = ' (4) ' (5) = 2 4 = 8.
If you don’t want to write them as powers of 2, you can read them o¤
from the powers of 2 in the last exercise: 2, 8, 3, 12, 2, 8, 3, 12.
Compare to Exercise 8 of 5.2 above.
Raji 5.4, Introduction to quadratic residues and nonresidues:
1. Find all quadratic residues of 3.
The nonzero squares modulo 3 are 12 = 1 and 22 = 4
quadratic residue of 3.
1
1, so 1 is the only
2. Find all quadratic residues of 13.
Again, we simply list the square 12 = 1, 22 = 4, 32 = 9, 42 = 3, 52 = 12,
62 = 10. We can quit here because 7 = 6, 8 = 5, etc. So the quadratic
residues are 1, 3, 4, 9, 10, and 12. Half of the twelve nonzero residues
modulo 13.
3. Find all quadratic residues of 18.
2
2
12 = 1, 52 = 7, 72 = 13, 112 = ( 7) = 13, 132 = ( 5) = 7, 172 =
2
( 1) = 1. So there are three of them: 1, 7, and 13.
2