Galois Theory - irreducibility over Q

  • Thread starter Kate2010
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In summary, Kate2010 is seeking help with a problem that involves showing that a polynomial is irreducible over Q for all n ≥ 2 given that a>1 is a product of distinct primes. The rational root theorem and Eisenstein's criterion are suggested as possible methods to approach the problem. After further discussion, it is determined that Eisenstein's criterion is applicable and can be used to solve the problem.
  • #1
Kate2010
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Homework Statement



If a>1 is a product of distinct primes, show that xn-a is irreducible over Q for all n ≥ 2.

Homework Equations





The Attempt at a Solution



I am not really sure how to start this problem. Can anyone point me in the right direction?

I know tests for irreducibility for example Eisensteins Criterion or reduction modulo p but I don't think that these are helpful here?

Thanks for any help.
 
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  • #2
Hi Kate2010! :smile:

How about the rational root theorem?
 
  • #3
I like Serena said:
Hi Kate2010! :smile:

How about the rational root theorem?

That would only tell you it doesn't have any linear factors. I'm a little confused why Kate2010 thinks Eisenstein's criterion isn't applicable.
 
  • #4
Dick said:
That would only tell you it doesn't have any linear factors. I'm a little confused why Kate2010 thinks Eisenstein's criterion isn't applicable.

Right.
Just looked up Eisenstein's criterion.
Looks like a good one. :)
 
  • #5
Thanks guys - I was trying to make things more complicated than they were.
 

FAQ: Galois Theory - irreducibility over Q

What is Galois Theory?

Galois Theory is a branch of mathematics that studies the properties of field extensions, which are algebraic structures that extend the set of rational numbers (Q) to include solutions to polynomial equations. It provides a powerful tool for understanding the solvability of polynomials and the symmetries of their solutions.

What does it mean for a polynomial to be irreducible over Q?

A polynomial is irreducible over Q if it cannot be factored into non-constant polynomials with rational coefficients. In other words, it cannot be broken down into simpler factors that also have rational coefficients. This property is important in Galois Theory because it helps us determine whether a polynomial is solvable using radicals.

How is irreducibility over Q related to the Galois group of a polynomial?

The Galois group of a polynomial is the group of symmetries of its roots, which can be thought of as the different ways the roots can be rearranged while still preserving the structure of the polynomial. The irreducibility of a polynomial over Q is closely linked to the size and structure of its Galois group. In particular, a polynomial is solvable by radicals if and only if its Galois group is a solvable group.

Can all polynomials with rational coefficients be solved using radicals?

No, not all polynomials with rational coefficients can be solved using radicals. This is because there are certain polynomials that are not solvable by radicals, such as the general quintic equation (polynomial of degree 5). Galois Theory helps us identify which polynomials can and cannot be solved using radicals by looking at the properties of their Galois groups.

How does Galois Theory apply to real-world problems?

Galois Theory has many applications in various fields, including physics, cryptography, and computer science. For example, it is used in error-correcting codes to efficiently transmit data over noisy channels, and in cryptography to ensure the security of data encryption algorithms. It also has applications in physics, specifically in the study of symmetries in quantum mechanics and the construction of mathematical models for physical phenomena.

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