Expert: Paul Klarreich - 5/29/2008

Question
QUESTION: Hi,

I am a first year algebra student and my teacher gave us challenge question. We are able to use what ever means to get the answer and she still doesn't think we will get it right. I have searched the internet on the topic but the explainations seem a little complicated for me to understand. I need it to be simplified so I can understand it.

We were told to state Eisenstein's Criteria for irreducibility of polynomials

Use the criteria to show that F(x)=x^5+4x^3+20x+2 is irreducible

And state one other irreducibility criteria.



Thank you for your help!

ANSWER: Questioner:   Donna
Category:  Advanced Math
Private:  No
 
Subject:  Eisenstein's Criteria for the irreduciblity of a polynomial
Question:  Hi,

I am a first year algebra student and my teacher gave us challenge question. We are able to use what ever means to get the answer and she still doesn't think we will get it right. I have searched the internet on the topic but the explainations seem a little complicated for me to understand. I need it to be simplified so I can understand it.

We were told to state Eisenstein's Criteria for irreducibility of polynomials

Use the criteria to show that F(x)=x^5+4x^3+20x+2 is irreducible

And state one other irreducibility criteria.
.....................................
Hi, Donna,

Yes, if you are not accustomed to this stuff, the explanations may seem complicated.

I am sure you found this one: [From Wikipedia -- I fixed up the notation a bit.]
...............................

In mathematics, Eisenstein's criterion gives sufficient conditions for a polynomial to be irreducible over the rational numbers (or equivalently, over the integers; see Gauss's lemma).

Suppose we have the following polynomial with integer coefficients.

an x^n + a[n-1]x^n-1 +... + a1x + a0


Assume that there exists a prime p such that

p divides each ai for a[n-1],.. down to a0, but
p does not divide an, and
p^2 does not divide a0.
Then f(x) is irreducible.
..............................
It says: Look for a prime that divides all coefficients but the leading one.  If you find one, see if its square, p^2, does NOT divide the last (constant) term.  If all that is so, the polynomial is irreducible (cannot be factored).
Your example is:

F(x) =  x^5 + 4x^3 + 20x + 2

Take p = 2.  The coefficients (except the first) are 0, 4, 20, 2.  p=2 divides them all. (see note)  But 2^2 = 4 does not divide the last one, a0 = 2.  So you have irreducibility.

Note:  Yes, Donna,  2 divides 0.  0 divided by 2 is zero, and it is perfectly legit.  You cannot use zero as a DIVISOR, but you are not doing that here.

About the last item -- sorry, I don't know any.


---------- FOLLOW-UP ----------

QUESTION: Mr. Klarreich,

Thanks for the information.  Can you please explain how zero is a coefficient in my example problem? I thought the coefficients were 1,4,20 and 2.

Answer
Hi, Donna,

In your example:

F(x) =  x^5 + 4x^3 + 20x + 2

you have an x^5 term but no x^4 term.  Since you are supposed to have all the terms from '5' down to '0', you must assign a coefficient of zero to any missing terms.

You will need to remember this in many situations.

Such as:

What is the slope of the line  y = 6?  The slope is the coefficient of x in the form  y = mx + b, right?  The coefficient of x is zero, so that is your slope.

Does that do it for you?