How Does Galois Theory Help Determine the Structure of the Dihedral Group D4?

Q is E=Q(4th root of 2, i) and G is the Galois group of E over Q. The minimal polynomial of the 4th root of 2 over Q and Q(i) is x^4-2 and the roots are +/-w, +/-wi where w is the 4th root of 2. The goal is to show that the Galois group H of E over Q(i) is a normal subgroup of G and that if K is the Galois group of Q(i) over Q, then it is isomorphic to G/H. Ultimately, this will prove that G is the group of symmetries, D4.
  • #1
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E=Q(4th root of 2, i) and G is the galios group of E over Q

I found the minimal polynomial p(x) of 4th root of 2 over Q and Q(i) to be
x^4-2

I'm trying to show

(1) the galios group H of E over Q(i) is a normal subgroup of G

(2) If K is the galios group of Q(i) over Q show that it is isomorphic to G/H

so I can ultimately show that G is actually D4 (the group of symmetries)

but I'm compeltely stuck
 
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  • #2
Okay, what have you done so far? What are the roots of the polynomial [itex]x^4= 2[/itex]? What is G? What is H?

By the way- it is 'Galois theory'. Capital G because it is a person's name and o before i.
 
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  • #3
HallsofIvy said:
Okay, what have you done so far? What are the roots of the polynomial [itex]x^4= 2[/itex]? What is G? What is H?

I found the minimal polynomial of 4th root of 2 over Q and Q(i) to be
x^4-2

and the roots are +/-w, +/-wi where w is the 4th root of 2
 
  • #4
Additional hint: What is the splitting field of [itex]x^4 - 2[/itex] over Q?

Petek
 

FAQ: How Does Galois Theory Help Determine the Structure of the Dihedral Group D4?

What is Galois theory?

Galois theory is a branch of abstract algebra that studies finite field extensions and their automorphisms. It was developed by French mathematician Évariste Galois in the 19th century.

Why is Galois theory important?

Galois theory is important because it provides a way to understand and classify finite field extensions, which have applications in many areas of mathematics and beyond. It also has connections to other branches of mathematics, such as number theory and group theory.

What are the main concepts in Galois theory?

The main concepts in Galois theory include fields, field extensions, automorphisms, and Galois groups. Fields are algebraic structures that satisfy certain properties, while field extensions are fields that contain another field as a subset. Automorphisms are bijective maps that preserve the structure of a field, and Galois groups are groups that correspond to the automorphisms of a field extension.

How is Galois theory applied in other fields?

Galois theory has applications in many areas of mathematics, such as algebraic number theory, algebraic geometry, and cryptography. It also has applications in physics, particularly in the study of symmetry and symmetry breaking.

What are some open problems in Galois theory?

Some open problems in Galois theory include the inverse Galois problem, which asks whether every finite group can be realized as the Galois group of a polynomial over the rational numbers. Other open problems include the exploration of certain types of field extensions, such as infinite or noncommutative extensions, and the study of the Galois theory of differential fields.

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