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

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SUMMARY

The discussion centers on determining the irreducibility of the polynomial f(x) = x^5 - 5x + 1 over the field Q(√-51)[x]. It is established that the Galois group of f(x) over the rationals is S_5, indicating that f(x) is irreducible over Q. The conversation also explores the existence of a third-degree irreducible polynomial in Q[x] that has a root in the splitting field M of f(x). The analysis utilizes the tower law and the fundamental theorem of Galois theory to conclude that if f reduces in K[x], it contradicts the properties of the Galois group.

PREREQUISITES
  • Understanding of Galois theory and Galois groups, specifically S_5.
  • Familiarity with irreducibility criteria for polynomials over fields.
  • Knowledge of the tower law in field extensions.
  • Basic concepts of splitting fields and their properties.
NEXT STEPS
  • Study the properties of Galois groups, focusing on S_5 and its implications for polynomial irreducibility.
  • Learn about the tower law and its applications in field extensions.
  • Explore the fundamental theorem of Galois theory and its relevance to polynomial roots.
  • Investigate examples of irreducible polynomials in Q[x] and their splitting fields.
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Mathematicians, algebraists, and students studying Galois theory, particularly those interested in polynomial irreducibility and field extensions.

peteryellow
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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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(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.)
 

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