Solved

Irreducible polynomials

Posted on 2004-08-25
3
492 Views
Last Modified: 2008-02-01
May I ask what are Irreducible polynomials?
Examples with workings will be good.

hongjun
0
Comment
Question by:hongjun
3 Comments
 
LVL 33

Assisted Solution

by:snoyes_jw
snoyes_jw earned 100 total points
ID: 11894451
Those polynomials that cannot be expressed as a product of non-trivial factors.  For example, x²-2 is irreducible over the set of rational numbers, because there are no rational numbers A and B such that x²-2 = (x+A)(x+B).

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
0
 
LVL 84

Assisted Solution

by:ozo
ozo earned 150 total points
ID: 11894506
A polynomial f(x) is irreducible over <R> if f(x) cannot be factoored as a product of polynomials in <R>[x] of degree less than the degree of f(x)
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)
0
 
LVL 4

Accepted Solution

by:
n_fortynine earned 250 total points
ID: 11918917
There is also a theorem that states that an irreducible polynimal p(x) in F[x] (where F is a field) will be reducible, i.e. have a root, in the extension field F[x]/p(x) (i.e. the field that contains all the remainders of a division of any polynomial in F[x] by p(x)). This theorem might not hold if F isn't a field.

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.
0

Featured Post

Industry Leaders: We Want Your Opinion!

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
Functions 7 83
Table function 6 56
Math homework question 5 100
Coordinate Geometry-Finding ratio of a point splitting a line 4 84
A Guide to the PMT, FV, IPMT and PPMT Functions In MS Excel we have the PMT, FV, IPMT and PPMT functions, which do a fantastic job for interest rate calculations.  But what if you don't have Excel ? This article is for programmers looking to re…
When we purchase storage, we typically are advertised storage of 500GB, 1TB, 2TB and so on. However, when you actually install it into your computer, your 500GB HDD will actually show up as 465GB. Why? It has to do with the way people and computers…
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201…

680 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question