QUADRATIC RECIPROCITY 73

possible to do a lot of mathematics that would otherwise be tedious

or too complicated to comprehend when stated in words.

Another example is powers. We write x

2

to mean x times x,

without g iving it a second thought. But as we write this, we can

think of the exponent as another variable, and write x

n

.Wecan

write down the power law of exponents: x

n

x

m

= x

n+m

. Inﬂuenced by

this symbolism, we can then start thinking about x

n

where n is no

longer an integer. We can write down x

1/2

and give a meaning to

that symbol: x

1/2

means the square root of x.

A fascinating example of the advantages and disadvantages

of notation occurred when calculus was invented (or discovered).

Newton used one symbol to stand for the process of differentiation;

Leibniz used a different symbol. Newton w as perhaps the greater

mathematician and scientist, but his notation was quite hard to

use. Leibniz’s notation was so good that every year, thousands

of students compute derivatives without ever fully understanding

them, because the notation almost forces the students to do the

right computation—at least most of the time!

We will see immediately that Legendre’s notation cleverly helps

us to notice and remember a certain multiplicative structure more

easily than we would have with the na

¨

ıve “unshifted” notation.

Multiplicativity of the Legendre Symbol

Legendre’s notation helps clarify what happens when p is ﬁxed and

the number a varies. In particular, after playing around with some

examples, you may discover—and it is possible to prove—that

a

p

b

p

=

ab

p

. (7.1)

EXERCISE: Show that

2

7

5

7

=

10

7

.

SOLUTION: You can check that 3

2

− 2 ≡ 0(mod7)and

4

2

− 2 ≡ 0 (mod 7). Therefore,

2

7

= 1. If you try the seven

Start Free Trial

No credit card required