Irreducible Polynomials

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In summary, we have discussed irreducible polynomials of degree less than or equal to 3 in Z2[x] and found the following: deg 1: x, x+1; deg 2: x^2+x+1; deg 3: x^3+x^2 + 1, x^3 + x +1. We also considered the polynomial f(x) = x4 + x + 1 and showed that it is irreducible over Z2. Lastly, we discussed factoring g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x], but found it to be a bit confusing. We also determined that x2 + 1 is not
  • #1
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(a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x].
(b) Show that f(x) = x4 + x + 1 is irreducible over Z2.
(c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x].


We have an irreducible polynomial if it cannot be factored into a product of polynomials of lower degree.
a)deg 1: x, x+1
deg 2: x^2+x+1
deg 3: x^3+x^2 + 1, x^3 + x +1
b) and c) get me confused.
I know a polynomial in F[x] is irreduble over F iff for all f(x),g(x) in F[x], p(x)|f(x)g(x) implies p(x)|f(x) or p(x)|g(x).
I don't know if that helps or if there is a simpler way to do this with degrees or something
 
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  • #2
kathrynag said:
(a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x].
(b) Show that f(x) = x4 + x + 1 is irreducible over Z2.
(c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x].

We have an irreducible polynomial if it cannot be factored into a product of polynomials of lower degree.
a)deg 1: x, x+1
deg 2: x^2+x+1
[STRIKE]How about x2 + 1 ?[/STRIKE] (See eumyang's post.)
deg 3: x^3+x^2 + 1, x^3 + x +1
b) and c) get me confused.
For (b): You have all of the irreducible polynomials of degree less than or equal to 3 in Z2[x]. Show that none of them divides f(x) = x4 + x + 1 .

For (c): Find one of the polynomials of degree less that 4, which divides g(x) = x5 + x + 1 .
I know a polynomial in F[x] is irreducible over F iff for all f(x),g(x) in F[x], p(x)|f(x)g(x) implies p(x)|f(x) or p(x)|g(x).
I don't know if that helps or if there is a simpler way to do this with degrees or something
I'm assuming the Z2 means Z2, the integers mod 2.
 
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  • #3
SammyS said:
How about x2 + 1 ?
That is not irreducible in Z2. x2 + 1 = (x + 1)2.
 
  • #4
Please delete this post.
 

1. What is an irreducible polynomial?

An irreducible polynomial is a polynomial that cannot be factored into smaller polynomials with coefficients from the same field.

2. How can I determine if a polynomial is irreducible?

There are several methods for determining if a polynomial is irreducible, including the rational root test, Eisenstein's criterion, and the reduction mod p test. These methods involve checking for certain patterns in the polynomial's coefficients.

3. Are all polynomials irreducible?

No, not all polynomials are irreducible. Some polynomials can be factored into smaller polynomials, while others, known as irreducible polynomials, cannot be factored any further.

4. What applications do irreducible polynomials have?

Irreducible polynomials are used in many areas of mathematics, including algebraic number theory, coding theory, and cryptography. They are also important in computer science for the construction of finite fields.

5. Can irreducible polynomials have complex coefficients?

Yes, irreducible polynomials can have complex coefficients. In fact, many important irreducible polynomials, such as the cyclotomic polynomials, have complex coefficients.

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