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Posted on 2004-08-25

May I ask what are Irreducible polynomials?

Examples with workings will be good.

hongjun

Examples with workings will be good.

hongjun

3 Comments

Straight from Google:

http://mathworld.wolfram.com/IrreduciblePolynomial.html

http://www.math.niu.edu/~beachy/aaol/polynomials.html

http://en.wikipedia.org/wiki/Irreducible_polynomial

for example, the polynomial x²+1 is

irreducible in the Reals, because x²+1 has no Real root

reducible in the Complex field because x²+1 = (x-i)(x+i)

reducible in Z2 because x²+1 = (x+1)²

reducuble in Z5 because x²+1 = (x + 3)(x+2)

A quick trick to recognize irreducibles of 2nd and 3rd degrees in F[x] is when they have no roots in F (F denotes a field).

For example: x^4 + x + 1 is irreducible in Z2[x], but has the root [x] in Z2[x]/(x^4 + x + 1) because [x]^4 + [x] + 1 = [x^4 + x + 1] = [0]

x^2 + x + 1 is also irreducible in Z2[x] but has the root [x^2 + x] in Z2[x]/(x^4 + x + 1) because [x^2 + x]^2 + [x^2 + x] + 1 = [x^4 + x + 1] = [0]

If you're unfamilar with rings, Z2 is the ring containing two elements [0] and [1], etc.

Hope this helps.

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