Determining Irreducibility of f(x) and Third Degree Polynomials in Q[x]

In summary, the conversation is about determining the irreducibility of a polynomial over a given field and whether there exists a third degree irreducible polynomial with a root in the splitting field of the given polynomial. The conversation also discusses using the tower law and fundamental theorem of Galois theory to show that the Galois group of the polynomial is not S_5.
  • #1
peteryellow
47
0
Please somebody help me with this it is very urgent.

I have that f(x) = x^5-5x+1 has S_5 as galois group over rationals. ANd M is the splitting field of f(x) over rationals.

Then how can I show that :

determine f(x) is irreducible over Q({-51}^{1/2})[x] or not?
Determine if there is third degree irreducible polynomial in Q[x], which has a root in M.
 
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  • #2
(a) Let Q({-51}^{1/2}) = K. If f reduces in K[x], what has to be true about the degree of the roots of f over Q? (Use the tower law.) In particular, what is [K : Q]? Why does this contradict the fact that the Galois group of f is S_5?

(b) This is the same thing as asking if there is a subfield of M of degree 3 over Q. (Hint: use the fundamental theorem of Galois theory.)
 

What does it mean for a polynomial to be irreducible?

For a polynomial to be irreducible, it means that it cannot be factored into a product of two polynomials with lower degrees. In other words, it cannot be broken down into simpler factors.

How do you determine if a polynomial is irreducible in Q[x]?

To determine if a polynomial is irreducible in Q[x], we can use the rational root theorem and check if there are any rational roots. If there are no rational roots, we can use the Eisenstein's criterion or the reducibility criterion to see if the polynomial can be factored into lower degree polynomials.

What is the difference between irreducible and prime polynomials?

Irreducible polynomials cannot be factored into lower degree polynomials, while prime polynomials cannot be factored into two non-constant polynomials. In other words, all prime polynomials are irreducible, but not all irreducible polynomials are prime.

Can a third degree polynomial always be factored into linear factors?

No, not all third degree polynomials can be factored into linear factors. This is because a third degree polynomial may have complex roots, which cannot be expressed as linear factors. However, if the polynomial has all rational roots, it can be factored into linear factors.

Is it possible for a polynomial to be irreducible in one field but reducible in another?

Yes, it is possible for a polynomial to be irreducible in one field, such as Q[x], but reducible in another field, such as R[x]. This is because different fields have different sets of elements, which can affect the factors of a polynomial.

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