*In which we study the structure of the multiplicative group and start thinking about quadratic residues.*

- Theorem 18:
*Let be a prime and let be a natural number. Then is cyclic.*We proved that the given by Lemma 17 has order modulo , for each . - Definition of
*quadratic residues*and*quadratic non-residues*. - Lemma 19:
*Let be an odd prime. Then there are exactly quadratic residues modulo (and so quadratic non-residues).*We gave two proofs of this. For the first, we thought about when we have . For the second, we thought about which powers of a primitive root are quadratic residues. Both proofs have ideas that are useful elsewhere. - Definition of the
*Legendre**symbol*.

#### Understanding today’s lecture

I suggested three exercises during the lecture.

What is the structure of ?

For which is cyclic?

If is a primitive root modulo , must it also be a primitive root modulo ?

You could also practise evaluating some Legendre symbols (with actual numbers).

#### Further reading

Baker (*A concise introduction to the theory of numbers*) deals with some of the exercises I suggested above and the material from today’s lecture in an extremely concise way; Jones and Jones (*Elementary Number Theory*) explain it in gory detail, with examples. Take your pick.

#### Preparation for Lecture 6

We know from Fermat’s Little Theorem that if is prime and is coprime to then . Why does that mean that ? Which give and which give ?

For which primes is a quadratic residue?

For which primes is a quadratic residue?

October 23, 2013 at 10:51 am

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