Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Polynomial greatest common divisor wikipedia , lookup
Compressed sensing wikipedia , lookup
Capelli's identity wikipedia , lookup
Root of unity wikipedia , lookup
Perron–Frobenius theorem wikipedia , lookup
Modular representation theory wikipedia , lookup
Deligne–Lusztig theory wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Algebraic number field wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
irreducibility of binomials with unity coefficients∗ pahio† 2013-03-21 19:06:25 Let n be a positive integer. We consider the possible factorization of the binomial xn +1. • If n has no odd prime factors, then the binomial xn + 1 is irreducible. Thus, x+1, x2 +1, x4 +1, x8 +1 and so on are irreducible polynomials (i.e. 2 irreducible in the field Q of their coefficients). N.B., only √ x+1 and x√+1 are irreducible in the field R; e.g. one has x4+1 = (x2−x 2+1)(x2+x 2+1). • If n is an odd number, then xn +1 is always divisible by x+1: xn + 1 = (x + 1)(xn−1 − xn−2 + xn−3 − + · · · − x + 1) (1) This formula is usable when n is an odd prime number, e.g. x5 + 1 = (x + 1)(x4 − x3 + x2 − x + 1). • When n is not a prime number but has an odd prime factor p, say n = mp, then we write xn +1 = (xm )p +1 and apply the idea of (1); for example: x12 + 1 = (x4 )3 + 1 = (x4 + 1)[(x4 )2 − x4 + 1] = (x4 + 1)(x8 − x4 + 1) There are similar results for the binomial xn + y n , and the formula corresponding to (1) is xn + y n = (x + y)(xn−1 − xn−2 y + xn−3 y 2 − + · · · − xy n−2 + y n ), (2) which may be verified by performing the multiplication on the right hand side. ∗ hIrreducibilityOfBinomialsWithUnityCoefficientsi created: h2013-03-21i by: hpahioi version: h36982i Privacy setting: h1i hResulti h12D05i h13F15i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1