Applying FTGT to Q(√2, √3)/Q

Abstract Algebra · Axiom Academy

Complete correspondence between the Klein four-group and intermediate fields Excellent! You've mastered the Fundamental Theorem of Galois Theory applied to ( , )/ . Perfect Correspondence: There's a one-to-one correspondence between subgroups of the Galois group and intermediate fields, reversing inclusion. Degree-Index Formula: For any subgroup H , we have [K^H : ] = |G : H| . Larger subgroups correspond to smaller fields! Klein Four-Group Structure: The Galois group ( ( , )/ ) /2 /2 has exactly 5 subgroups, giving exactly 5 intermediate fields. Finding Fixed Fields: To find the fixed field of a subgroup H , look for elements fixed by all automorphisms in H . For example, fixes , so ( ) is contained in the fixed field of . The Power of FTGT: This theorem transforms questions about field extensions into questions about group theory, and vice versa! This correspondence is the heart of Galois theory. You can now apply FTGT to analyze any Galois extension by studying its Galois group's subgroup lattice!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.