Solvable Group: Need Help Understanding Fraleigh's Exercise

In summary, the conversation discusses the concept of a solvable group in abstract algebra and its connection to finite Galois extensions of finite fields. It is stated that the Galois group of a finite extension of a finite field is solvable, and this is true because finite Galois extensions of finite fields have cyclic Galois groups, which are solvable. The term "solvable by radicals" is also mentioned and connected to the solvability of polynomials. The use of the term "cyclic" is deduced by the primitive element theorem.
  • #1
emptyboat
28
1
Hello...

I have a question about the solvable group.

I read a Fraleigh's 'A first course in abstract algebra', there is a question in sec56, exercise3.

It says "The Galois group of a finite extension of a finite field is solvable." is true...

I can't figure out why it's true.

I think this means "every finite polynomial of finite field F is solvable by radicals." (it's correct?)

Help...
 
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  • #2
A finite Galois extension of a finite field has cyclic Galois group, and any cyclic group is solvable.

You are correct in thinking that the term is connected to the solvability of polynomials by radicals: a polynomial is solvable by radicals if and only if it has a solvable galois group.
 
Last edited:
  • #3
Thanks a lot, mrbohn1.

I understand it.

I think 'Cyclic' is deduced by 'primitive element Theorem'.
 

1. What is a solvable group?

A solvable group is a type of group in abstract algebra that can be built up from simpler groups by taking successive quotients. This means that every subgroup of the group can be reached by taking the quotient of a previous subgroup. It is a useful concept in understanding the structure and properties of certain groups.

2. How do I determine if a group is solvable?

To determine if a group is solvable, you can use the derived series method. This involves taking the commutator subgroup of the group, and continuing to take the commutator subgroup of the previous subgroup until you reach the identity element. If the final subgroup is the identity element, then the group is solvable. If not, then the group is not solvable.

3. What is the significance of solvable groups?

Solvable groups have many important applications in mathematics, particularly in the study of Galois theory and the construction of certain finite groups. They also have connections to other areas of mathematics such as number theory and geometry. Understanding solvable groups can help in solving complex mathematical problems and developing new theories.

4. Can non-abelian groups be solvable?

Yes, non-abelian groups can be solvable. In fact, the majority of finite groups are solvable. Non-abelian groups that are solvable have a more complicated structure than abelian groups, as they involve both commutators and non-commutative operations. However, the concept of solvability still applies to these groups.

5. How can I use solvable groups to solve problems in abstract algebra?

Solvable groups can be used as a tool to simplify and understand complex groups. By breaking down a group into its simpler quotients, it becomes easier to analyze and make conclusions about its structure and properties. This can be useful in solving problems and proving theorems in abstract algebra. Solvable groups can also provide insight into the behavior of more complicated groups, allowing for further applications and developments in the field.

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