Lecture: Primitive Element Theorem and a Deterministic Primality Test

Background Material

• Euler's criterion from Section 6.2 of the text.

Reading

• Chapter 9 of the text.

Topics

1. Define order and illustrate primitive element theorem.
2. Theorem 9.1 and corollaries. If b^j = 1 (mod n), then j|ord(b).
3. Theorem 9.4. ord(b^i) = e/gcd(e,i), e = ord(b).
4. Proof of primitive element theorem.
5. Structure theorem for Z_n^* (units in Z_n).
6. A test to determine if a number is a primitive element (Lucas).
7. Brillhart, Lehmer, Selfridge primality test.

Maple Worksheet

1. selfridge.mw - Maple worksheet implementing the Brillhard, Lehmer, Selfridge primality test.

Practice Assignment

1. Exc. 9.10 and 9.11.