Irreducibility of a polynomial by eisenstein and substitution

In summary, the conversation discusses the use of Eisenstein's criteria to determine if a polynomial is irreducible. The conversation also considers the impact of making substitutions on the reducibility of a polynomial. The possibility of using linear substitutions in general is also discussed. It is suggested that a theory dealing with substitutions in an abstract sense may exist.
  • #1
icantadd
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0

Homework Statement


Show [tex]16x^4 = 8x^3 - 16x^2 - 8x + 1[/tex] is irreducible.


Homework Equations



Eisenstein's criteria, if there is n s.t. n does not divide the leading coefficient, divides all the other coefficients, and n^2 does not divide the last coefficient then the polynomial is irreducible (over the rationals)

The Attempt at a Solution


I want to say that consider [tex]p'(x) = p(\frac{1}{2}x) = x^4 + x^3 - 4x^2 - 4x + 1 [/tex] is irreducible by eisenstein if we use the standard trick of substituting x+1 -> x, then we get, [tex]x^4+5x^3+5x^2-5x-5[/tex] where eisenstein is immediate. What I don't know is that if I can then say that since p' is irreducible, then p is.
 
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  • #2
icantadd said:

Homework Statement


Show [tex]16x^4 = 8x^3 - 16x^2 - 8x + 1[/tex] is irreducible.
This should be [tex]16x^2+ 8x^4- 16x^2- 8x+ 1[/itex]


Homework Equations



Eisenstein's criteria, if there is n s.t. n does not divide the leading coefficient, divides all the other coefficients, and n^2 does not divide the last coefficient then the polynomial is irreducible (over the rationals)

The Attempt at a Solution


I want to say that consider [tex]p'(x) = p(\frac{1}{2}x) = x^4 + x^3 - 4x^2 - 4x + 1 [/tex] is irreducible by eisenstein if we use the standard trick of substituting x+1 -> x, then we get, [tex]x^4+5x^3+5x^2-5x-5[/tex] where eisenstein is immediate. What I don't know is that if I can then say that since p' is irreducible, then p is.
p(x) is reducible if and only if p(x)= a(x)b(x) for polynomials a and b of lower degree than p. In that case p(x/2)= a(x/2)b(x/2) so, yes, if p(x) is reducible so is p(x/2). And, then, if p(x/2) is irreducible, so is p(x).
 
  • #3
First of all, thank you for your help!

That makes sense. I was stuck because I had a counter example that in general one cannot make substitutions; for example, p(u) = u^2 + 8u + 36 is irreducible over Q, but under the substitution, [t^2 -> u] p(t) splits into quadratic factors. So perhaps, is it possible to claim that any linear substitution over the single variable will be acceptable (i.e. [a*x + b -> x])?

Is there a theory that deals with substitutions in the abstract sense?
 

1. What is the Eisenstein criterion for irreducibility of a polynomial?

The Eisenstein criterion is a test used to determine whether a polynomial with integer coefficients is irreducible over the rational numbers. It states that if a polynomial can be written in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0 and there exists a prime number p that divides all coefficients except an and a0 and p2 does not divide a0, then the polynomial is irreducible.

2. What is the substitution method for testing irreducibility of a polynomial?

The substitution method is another test used to determine whether a polynomial is irreducible over the rational numbers. It involves substituting a suitable value for x in the polynomial and checking if the resulting polynomial is irreducible. If it is, then the original polynomial is also irreducible.

3. Can a polynomial be irreducible using both the Eisenstein and substitution criteria?

Yes, it is possible for a polynomial to satisfy both the Eisenstein and substitution criteria for irreducibility. This means that the polynomial is irreducible over the rational numbers and cannot be factored into polynomials with integer coefficients.

4. What is the importance of the irreducibility of a polynomial?

The irreducibility of a polynomial is an important concept in algebra and number theory. It allows us to determine whether a polynomial can be factored into simpler polynomials, and helps in solving equations and finding roots. It is also used in other areas of mathematics, such as cryptography and coding theory.

5. Are there any limitations to the Eisenstein and substitution criteria for irreducibility?

Yes, there are limitations to both the Eisenstein and substitution criteria. The Eisenstein criterion can only be applied to polynomials with integer coefficients, while the substitution method may not always work for certain polynomials. In some cases, more advanced methods such as the rational root theorem or the use of complex numbers may be necessary to determine irreducibility.

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