Constructing the Galois Group of f

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In summary, the permutation group that is isomorphic to the Klein4 group is generated by the e, (sqrt(-3),-sqrt(-3)),(sqrt(2),-sqrt(2)),(sqrt(2),-sqrt(2))(sqrt(-3),-sqrt(-3)) elements.
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Hey there,
firstly I hope that this is the right place to discuss such things. if not, could you direct me somewhere else?
Ok, I have to construct the Galois Group of f= (x^2-2x-1)^3 (x^2+x+1)^2 (x+1)^4 and then represent it as a permutation group of the roots.

first I constructed the splitting field extension S:Q (where S= summation symbol and Q = field of Rational numbers)

The splitting field i Came up with was Q(sqrt (2), sqrt (-3)):Q, and the degree of this splitting field is 4...am I correct here? is this the splitting field?

The Galois group represented as a permutation group I ended up getting was
{ e (the identity), (sqrt(-3),-sqrt(-3)),(sqrt(2),-sqrt(2)),(sqrt(2),-sqrt(2))(sqrt(-3),-sqrt(-3))}
isomorphic to the Klein4 group...
am i doing this right?? it just seems abit simple a result for an initial function that wasn't that simple ! :rolleyes:
 
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  • #2
The splitting field for a polynomial is the same as the splitting field for the product of its irreducible, non-linear factors. Thus the splitting field of f(x) is the same as that of (x^2-2x-1)(x^2+x+1).
 
  • #3
Excellent, thank you very much. the point you have made will help me in the future too ! Is my permutation group correct...not entirely sure as to whether this is all that's required!
 
  • #4
You know that every permutation must take a root of one polynomial to a root of the same polynomial, so the ones you've written are the only possible ones. However, it may be that not all of those are valid. Have you seen the connection between the degree of a splitting field and the size of its Galois group?
 
  • #5
Yes, the degree of the splitting field is 4, so there are 4 elements in the Galois Group too, so I think I'm done. Thank you for your advice!
 

FAQ: Constructing the Galois Group of f

What is the Galois Group of a polynomial?

The Galois Group of a polynomial, denoted as Gal(f), is a mathematical group that represents all the possible permutations of the roots of the polynomial f. It is used to study the symmetry and solvability of equations.

How do you construct the Galois Group of a polynomial?

The Galois Group of a polynomial can be constructed by first finding the roots of the polynomial, then determining the automorphisms (permutations) of these roots that leave the polynomial unchanged. These automorphisms are then combined to form the Galois Group.

What is the significance of the Galois Group in mathematics?

The Galois Group plays a crucial role in understanding the solvability of polynomial equations. It also has applications in other areas of mathematics, such as group theory, abstract algebra, and number theory.

Can the Galois Group of a polynomial be computed for any polynomial?

No, the Galois Group can only be computed for polynomials whose roots can be expressed in terms of radicals (square roots, cube roots, etc.). Polynomials with roots that cannot be expressed in this way, such as the roots of x^5 + x + 1, do not have a well-defined Galois Group.

Are there any practical applications of the Galois Group?

Yes, the Galois Group has practical applications in fields such as cryptography, coding theory, and computer science. It is also used in Galois theory, a branch of abstract algebra, to study the relationship between field extensions and field automorphisms.

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