special reducible polynomials over a field with positive characteristic

special reducible polynomials over a field with positive characteristic

Let

k

be an arbitrary field such that

char

⁢

(

k

)

=

p

>

0

. We will assume that

0

∉

ℕ

.

Proposition

. Let

m

∈

ℕ

. Then for any

a

∈

k

the

polynomial

W

⁢

(

X

)

=

X

p

m

-

a

is reducible if and only if there exist

c

∈

k

and

n

∈

ℕ

such that

c

p

n

=

a

. Moreover the factorization of

W

⁢

(

X

)

is given by the

formula

W

⁢

(

X

)

=

(

X

p

m

-

n

-

c

)

p

n

,

where

n

is a maximal

natural number

such that

0

≤

n

≤

m

and

a

=

c

p

n

for some

c

∈

k

.

Proof

. “

⇐

” Assume that

a

=

c

p

n

for some

c

∈

k

and

n

∈

ℕ

. It is well known that if

char

⁢

(

k

)

=

p

>

0

and

t

∈

ℕ

then for any

x

,

y

∈

k

we have

(

x

+

y

)

p

t

=

x

p

t

+

y

p

t

. Therefore

W

⁢

(

X

)

=

X

p

m

-

a

=

X

p

m

-

c

p

n

=

(

X

p

m

-

1

)

p

-

(

c

p

n

-

1

)

p

=

(

X

p

m

-

1

-

c

p

n

-

1

)

p

=

(

V

⁢

(

X

)

)

p

.

Note that

p

m

>

deg

⁢

(

V

⁢

(

X

)

)

=

p

m

-

1

>

0

and therefore

W

⁢

(

X

)

is reducible.

□

“

⇒

” Assume that

W

⁢

(

X

)

is reducible. Therefore there exist

V

⁢

(

X

)

,

U

⁢

(

X

)

∈

k

⁢

[

X

]

such that

W

⁢

(

X

)

=

V

⁢

(

X

)

⋅

U

⁢

(

X

)

and both

deg

⁢

(

V

⁢

(

X

)

)

>

0

and

deg

⁢

(

U

⁢

(

X

)

)

>

0

.

Recall that there exists an algebraically closed field

k

¯

such that

k

is a subfield of

k

¯

(generally it is true for any field). Therefore there exists

c

0

∈

k

¯

such that

c

0

p

m

=

a

and thus we have:

W

⁢

(

X

)

=

X

p

m

-

a

=

X

p

m

-

c

0

p

m

=

(

X

-

c

0

)

p

m

in

k

¯

⁢

[

X

]

. Now

V

⁢

(

X

)

⋅

U

⁢

(

X

)

=

W

⁢

(

X

)

=

(

X

-

c

0

)

p

m

and since

k

¯

⁢

[

X

]

is a

unique factorization domain

then for

n

=

deg

⁢

(

V

⁢

(

X

)

)

>

0

we have:

V

⁢

(

X

)

=

(

X

-

c

0

)

n

.

But

V

⁢

(

X

)

∈

k

⁢

[

X

]

(the factorization was assumed to be over

k

) and therefore

c

0

n

∈

k

. It is easy to see that since

c

0

n

∈

k

and

c

0

p

m

∈

k

then

c

0

gcd

⁢

(

n

,

p

m

)

∈

k

, but

gcd

⁢

(

n

,

p

m

)

=

p

s

for some

s

∈

ℕ

. Thus if we put

c

=

c

0

p

s

we gain that

c

p

m

-

s

=

a

. But

m

>

s

(since

n

<

p

m

because we assumed that both

deg

⁢

(

V

⁢

(

X

)

)

>

0

and

deg

⁢

(

U

⁢

(

X

)

)

>

0

), which

completes

the proof of the first part.

□

Now let

n

∈

ℕ

be a maximal natural number such that

n

≤

m

and

a

=

c

p

n

for some

c

∈

k

. Then we have

W

⁢

(

X

)

=

(

X

p

m

-

n

-

c

)

p

n

.

Note that the polynomial

X

p

m

-

n

-

c

is

irreducible

. Indeed, assume that

X

p

m

-

n

-

c

is reducible. Then (due to first part of the proposition)

c

=

u

p

k

for some

k

∈

ℕ

and

u

∈

k

. But then

a

=

(

u

p

k

)

p

n

=

u

p

n

+

k

.

Contradiction

, since

n

+

k

>

n

and

n

was assumed to be maximal.

□

Title

special reducible polynomials over a field with positive characteristic

Canonical name

SpecialReduciblePolynomialsOverAFieldWithPositiveCharacteristic

Date of creation

2013-03-22 18:31:05

Last modified on

2013-03-22 18:31:05

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

10

Author

joking (16130)

Entry type

Theorem

Classification

msc 13F07

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