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Irreducible Polynomial

A Polynomial or polynomial equation is said to be irreducible if it cannot be factored into polynomials of lower degree over the same Field.

The number of binary irreducible polynomials of degree $n$ is equal to the number of $n$-bead fixed Necklaces of two colors: 1, 2, 3, 4, 6, 8, 14, 20, 36, ... (Sloane's A000031), the first few of which are given in the following table.

$n$ Polynomials
1 $x$
2 $x$, $x+1$
3 $x$, $x^2+x+1$, $x+1$
4 $x$, $x^3+x+1$, $x^3+x^2+1$, $x+1$

See also Field, Galois Field, Necklace, Polynomial, Primitive Irreducible Polynomial


Sloane, N. J. A. Sequence A000031/M0564 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and extended entry in Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

© 1996-9 Eric W. Weisstein