Test Your Knowledge

Challenge yourself with questions covering all Galois theory concepts

Galois Theory Comprehensive Quiz

Test your mastery of Galois Theory! This comprehensive quiz covers all the major topics: field extensions, polynomial theory, the Galois correspondence, solvability theory, and real-world applications. With 30 carefully crafted questions spanning three difficulty levels, you'll demonstrate your understanding of one of mathematics' most beautiful theories.

Easy (10 questions)
Basic definitions, fundamental concepts, and straightforward calculations
Medium (12 questions)
Applications of theory, Galois groups, intermediate fields, and examples
Hard (8 questions)
Deep conceptual understanding, complex correspondences, and advanced topics

Topics Covered:

đź“š Foundations
Field automorphisms, extension degree, tower law, normal and separable extensions
🔢 Polynomials
Irreducibility, splitting fields, minimal polynomials, discriminants
đź”— Galois Correspondence
Fundamental theorem, subgroup-field correspondence, Galois groups of famous polynomials
âž• Field Extensions
Finite fields, algebraic closure, Frobenius automorphism, primitive elements
🔓 Solvability Theory
Solvable groups, radical extensions, unsolvable quintic, Abel-Ruffini theorem
🌍 Applications
Cyclotomic fields, geometric constructions, cryptography, error correction codes
Question 1 of 30
EASYFoundations

What is a field automorphism?

0 / 30 answered

How to Use This Quiz:

1.Start with easy questions to build confidence, then work your way up to harder ones. Use the difficulty filter to focus on one level at a time.
2.Focus on specific topics using the topic filter. If you're struggling with a particular area (like solvability or the Galois correspondence), practice just those questions.
3.Read the explanations carefully after submitting. Understanding why an answer is correct is more important than getting it right!
4.Retake the quiz after studying. Galois Theory is deep—it takes time and repetition to master. Aim for 90%+ on hard questions!
5.Use the interactive tools in the Playground to explore concepts you're unsure about. Hands-on experimentation builds intuition.
Scoring Guide:
• 90-100%: Outstanding! You've mastered Galois Theory.
• 70-89%: Great work! You understand most concepts. Review the questions you missed.
• 50-69%: Good foundation. Study the intermediate topics more deeply.
• Below 50%: Keep studying! Galois Theory is challenging. Go back through the demos and examples.

You've Reached the End!

Congratulations on making it through the entire Galois Theory module! From the basic definitions of field extensions to the profound insight that the unsolvable quintic is a consequence of Sâ‚… being non-solvable, you've journeyed through one of the most beautiful theories in all of mathematics.

What You've Learned:
âś“ Field theory and automorphisms
âś“ Polynomial irreducibility and splitting fields
âś“ The Fundamental Theorem of Galois Theory
âś“ Why there's no quintic formula
âś“ Real-world applications in cryptography and coding theory
Where to Go Next:
→ Study algebraic number theory
→ Explore representation theory
→ Learn about class field theory
→ Investigate algebraic geometry
→ Apply these ideas to cryptography or coding theory
Évariste Galois' Legacy
Galois died at age 20 in a duel in 1832, the night before frantically writing down his mathematical ideas. His work was so revolutionary that it took decades to be fully understood and appreciated. Today, Galois Theory stands as one of mathematics' greatest achievements—a perfect bridge between algebra, geometry, and group theory. It solved ancient problems, created new fields of study, and powers modern technology. His story reminds us that the most profound ideas often come from the most unexpected places, and that mathematics transcends time and circumstance.

Thank you for learning Galois Theory! Whether you're a student, a researcher, or just curious about mathematics, we hope this module has deepened your appreciation for the beauty and power of abstract algebra.