# p-adic Number

**Theorem:** If one has a solution of the form

x^{2} = b mod p,

with p > 2, then the solutions obtained when raising p to higher powers exist,

and will form a coherent sequence.

Proof: The proof will be by induction. The hypothesis of the theorem takes

care of the first part of the induction. Now assume we have
mod p^{n},

so for some
. We want to know if there exists a
such that

Expanding and replacing with we get

Cancelling out the b's and the term divisible by we get

Now, dividing by p ^{n} we get

We will now show that no matter what is, we can choose
so that

the equality is true. Assume = 2m (i.e., is even). Then let
= (p -

mod p. Assume = 2m + 1, then let . Of
course,

will exist because p is a prime. Therefore we will be able
to carry on the

sequence infinitely.

Even with just our one solution of x ^{2} = 2 (the 7-adic solution), we

have shown that there is at least one element in which is not in Q, and

therefore is strictly bigger than Q.

In what will at first seem to take us in a new direction, we define the

p-adic valuation on the rational numbers .

**Definition** The p-adic valuation of a non-zero integer n is the unique number

such that

We extend this definition to a rational number
by saying

Finally, we define .

Now we define the p-adic absolute value.

**Definition** For any x ∈ Q not equal to 0, we define the p-adic absolute value

of x to be

and if x = 0, we set .

We refer to that as an absolute value, but to ensure that it is an absolute

value we must verify the following three properties (which, if true, will show

that this absolute value is a non -archimedian absolute value.)

1. l xl = 0 if and only if x = 0.

2. for all x, y ∈ Q.

3.

To show this is true, we will rely on the following Lemma:

**
Lemma** For all x and y ∈ Q, we have

Proof: We begin by assuming that both x and y are
integers. Let

and , with both x' and y' not divisible by p . Now

and therefore . To prove the second
part of the

Lemma, we assume that a ≤ b. (We can always reverse the roles of x and y

if we need to.) With this in mind,

Therefore, : Now we must
make sure this

is also true when x is not an integer but a rational number . Assume

and , then:

And,

Now, assume that the . Then,

This concludes the proof of the Lemma.

We are now ready to show that the p-adic absolute value as we defined

it is, in fact, an absolute value and, more specifically, is a non-archimedian

absolute value.

**Theorem: **Our definition above defines a non-archimedian absolute

value on Q.

Proof:

1. For the first property , note that the absolute value cannot equal zero

for any finite . Therefore
if and only if x = 0.

2. We can prove the second property simply by using the Lemma and

manipulating the definitions.

3. Once again, we can prove this property directly from the Lemma.

This concluces the proof.

With more time it could be shown that the p-adic numbers dealt with in

the first half of the paper are a completion of Q with respect to the p-adic

absolute value dealt with in the second half of the paper.

**Acknowledgements:** I would like to thank Harvey Keynes for telling me

about this program, the University of Minnesota for paying me to attend

lectures and read math texts , and most of all Professor Paul Garrett for

giving me the opportunity to work here this summer and for his daily lectures

and assistance.

Created using.

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