formal congruence

formal congruence

Let

m

be a

positive integer

.  Two

polynomials

f

⁢

(

X

1

,

X

2

,

…

,

X

n

)

and

g

⁢

(

X

1

,

X

2

,

…

,

X

n

)

with

integer

coefficients are said to be

formally congruent

modulo

m

, denoted

f

⁢

(

X

1

,

X

2

,

…

,

X

n

)

⁢

≡

¯

⁢

g

⁢

(

X

1

,

X

2

,

…

,

X

n

)

(

mod

m

)

,

(1)

iff all coefficients of the difference polynomial

f

-

g

are

divisible

by

m

.

Remark 1.

The formal congruence of polynomials is an

equivalence relation

in the set

ℤ

⁢

[

X

1

,

X

2

,

…

,

X

n

]

.

Remark 2.

The formal congruence (1) implies that all integers substituted for the

variables

satisfy

it, in other words, one can speak of an

identical congruence

.  However, there are identical congruences that are not formal congruences; e.g.

X

p

≡

X

(

mod

p

)

where

p

is a

positive

prime.

Examples

1.

(

X

+

Y

)

p

⁢

≡

¯

⁢

X

p

+

Y

p

(

mod

p

)

when

p

is a positive

prime number

.  This result is easily proved with the

binomial theorem

(cp. the freshman’s dream) and may be by

induction

generalized to

(

X

1

+

X

2

+

…

+

X

n

)

p

⁢

≡

¯

⁢

X

1

p

+

X

2

p

+

…

+

X

n

p

(

mod

p

)

.

If one substitutes in this formal congruence 1 for all

X

1

,

X

2

, …,

X

n

, one obtains the

congruence

n

p

≡

n

(

mod

p

)

;

substitution of

-

1

shows that the last congruence is valid also for negative integers

n

.  If it is supposed that

p

is not factor of

n

, we have got the Fermat’s little

theorem

.

2.

Let

p

be a positive prime number.  It is possible that the

n

th

degree congruence

P

⁢

(

x

)

:=

a

0

⁢

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

≡

0

(

mod

p

)

,

where

a

i

∈

ℤ

⁢

∀

i

and

p

∤

a

0

,  has

n

modulo

p

incongruent roots

x

=

x

1

,

x

2

,

…

,

x

n

.  We then have the formal congruence

P

⁢

(

x

)

⁢

≡

¯

⁢

a

0

⁢

(

x

-

x

1

)

⁢

(

x

-

x

2

)

⁢

⋯

⁢

(

x

-

x

n

)

(

mod

p

)

.

Especially, we have

x

p

-

1

-

1

⁢

≡

¯

⁢

(

x

-

1

)

⁢

(

x

-

2

)

⁢

⋯

⁢

(

x

-

p

+

1

)

(

mod

p

)

,

because Fermat’s little theorem guarantees the roots

x

=

1

,

2

,

…

,

p

-

1

for the congruence

x

p

-

1

-

1

≡

0

(

mod

p

)

.  If the value

x

=

0

is substituted in the previous formal congruence, it gives

-

1

≡

(

-

1

)

p

-

1

⁢

(

p

-

1

)

!

(

mod

p

)

or

(

p

-

1

)

!

≡

-

1

(

mod

p

)

,

which is half of Wilson’s theorem.  The reverse part of this theorem follows from the last congruence so that if

p

were not prime, then the number

-

1

would be divisible by any

proper divisor

of

p

.

References

1

F. Stöwener

: “Simultanbeweis des Fermatschen und Wilsonschen Satzes”.  –

Elemente der Mathematik

30

(1975).

Title

formal congruence

Canonical

name

FormalCongruence

Date of creation

2013-03-22 14:23:50

Last modified on

2013-03-22 14:23:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

32

Author

pahio (2872)

Entry type

Definition

Classification

msc 11A07

Classification

msc 11C08

Synonym

polynomial congruence

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Related topic

IdealDecompositionInDedekindDomain

Related topic

ExampleOfGcd

Related topic

FermatsLittleTheorem

Related topic

WilsonsTheorem

Related topic

FreshmansDream

Related topic

UsingPrimitiveRootsAndIndexToSolveCongruences

Related topic

SufficientConditio

Defines

formally congruent

Defines

identical congruence

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