Solvable Group: Need Help Understanding Fraleigh's Exercise

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SUMMARY

The discussion centers on the solvability of the Galois group of finite extensions of finite fields as stated in Fraleigh's 'A First Course in Abstract Algebra'. It is established that every finite polynomial over a finite field is solvable by radicals, which is directly linked to the cyclic nature of the Galois group in this context. The participants confirm that a finite Galois extension of a finite field indeed has a cyclic Galois group, which is inherently solvable. The connection to the Primitive Element Theorem is also highlighted as a key concept in understanding this relationship.

PREREQUISITES
  • Understanding of Galois theory
  • Familiarity with finite fields
  • Knowledge of cyclic groups
  • Concept of solvability of polynomials by radicals
NEXT STEPS
  • Study the Primitive Element Theorem in detail
  • Explore the properties of cyclic groups in group theory
  • Learn about the implications of Galois theory on polynomial solvability
  • Examine examples of finite extensions of finite fields
USEFUL FOR

Students of abstract algebra, mathematicians interested in Galois theory, and anyone seeking to deepen their understanding of the relationship between finite fields and solvable groups.

emptyboat
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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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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:
Thanks a lot, mrbohn1.

I understand it.

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

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