prime factors of x^n-1

prime factors of

x

n

-

1

We list

prime factor

of the

binomials

x

n

-

1

in

ℚ

, i.e. in the

polynomial ring

ℚ

⁢

[

x

]

.  The prime factors can always be chosen to be with

integer

coefficients

and the number of the prime factors equals to

τ

⁢

(

n

)

(

http://planetmath.org/TauFunction

); see the proof (

http://planetmath.org/FactorsOfNAndXn1

).

x

-

1

x

2

-

1

=

(

x

+

1

)

⁢

(

x

-

1

)

x

3

-

1

=

(

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

4

-

1

=

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

5

-

1

=

(

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

6

-

1

=

(

x

2

+

x

+

1

)

⁢

(

x

2

-

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

7

-

1

=

(

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

8

-

1

=

(

x

4

+

1

)

⁢

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

9

-

1

=

(

x

6

+

x

3

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

10

-

1

=

(

x

4

+

x

3

+

x

2

+

1

)

⁢

(

x

4

-

x

3

+

x

2

-

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

11

-

1

=

(

x

10

+

x

9

+

x

8

+

x

7

+

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

12

-

1

=

(

x

4

-

x

2

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

2

-

x

+

1

)

⁢

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

13

-

1

=

(

x

12

+

x

11

+

x

10

+

x

9

+

x

8

+

x

7

+

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

14

-

1

=

(

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

6

-

x

5

+

x

4

-

x

3

+

x

2

-

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

15

-

1

=

(

x

8

-

x

7

+

x

5

-

x

4

+

x

3

-

x

+

1

)

⁢

(

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

16

-

1

=

(

x

8

+

1

)

⁢

(

x

4

+

1

)

⁢

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

17

-

1

=

(

x

16

+

x

15

+

x

14

+

…

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

18

-

1

=

(

x

6

+

x

3

+

1

)

⁢

(

x

6

-

x

3

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

2

-

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

19

-

1

=

(

x

18

+

x

17

+

x

16

+

…

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

20

-

1

=

(

x

8

-

x

6

+

x

4

-

x

2

+

1

)

⁢

(

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

4

-

x

3

+

x

2

-

x

+

1

)

⁢

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

21

-

1

=

(

x

12

-

x

11

+

x

9

-

x

8

+

x

6

-

x

4

+

x

3

-

x

+

1

)

⁢

(

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

22

-

1

=

(

x

10

+

x

9

+

x

8

+

x

7

+

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

)

⁢

(

x

10

-

x

9

+

x

8

-

x

7

+

x

6

-

x

5

+

x

4

-

x

3

+

x

2

-

x

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

x

23

-

1

=

(

x

22

+

x

21

+

x

20

+

…

+

x

2

+

x

+

1

)

⁢

(

x

-

1

)

x

24

-

1

=

(

x

8

-

x

4

+

1

)

⁢

(

x

4

-

x

2

+

1

)

⁢

(

x

4

+

1

)

⁢

(

x

2

+

x

+

1

)

⁢

(

x

2

-

x

+

1

)

⁢

(

x

2

+

1

)

⁢

(

x

+

1

)

⁢

(

x

-

1

)

Note 1.

All factors shown above are

irreducible polynomials

(in the field

ℚ

of their own coefficients), but of course they (except

x

±

1

) may be split into factors of

positive

degree in certain

extension fields

; so e.g.

x

4

+

1

=

(

x

2

+

x

⁢

2

+

1

)

⁢

(

x

2

-

x

⁢

2

+

1

)

in

⁢

the

⁢

field

⁢

ℚ

⁢

(

2

)

.

Note 2.

The 24 examples of

factorizations

are true also in the fields of

characteristic

≠

0

, but then many of the factors can be simplified or factored onwards (e.g.

x

2

+

1

≡

(

x

+

1

)

2

if the characteristic (

http://planetmath.org/Characteristic

) is 2).

Title

prime factors of

x

n

-

1

Canonical name

PrimeFactorsOfXn1

Date of creation

2013-03-22 16:29:51

Last modified on

2013-03-22 16:29:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Result

Classification

msc 13G05

Related topic

GausssLemmaII

Related topic

IrreducibilityOfBinomialsWithUnityCoefficients

Related topic

FactorsOfNAndXn1

Related topic

ExamplesOfCyclotomicPolynomials

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