The Galois Correspondence
The symmetries of a field extension form a group whose subgroups match the intermediate fields, in reverse order. Whether that group can be built from abelian pieces is what decides if the roots can be written with radicals.
What a learner can do afterwards
- List the automorphisms of a small splitting field
- Match subgroups to intermediate fields in a concrete degree-four example
- State what solvability of the group says about a general quintic
1 · Read
Start with a polynomial and adjoin all its roots at once. The result is called a splitting field. An automorphism is a relabeling of that field that fixes every rational number and preserves all arithmetic. Each automorphism permutes the roots, so the whole symmetry group acts as permutations of the roots. For a Galois extension, the number of automorphisms equals the degree of the extension.
Take the rationals with square roots of 2 and 3 adjoined. The total degree is 2 times 2, which is 4, so the group has order 4. It is the Klein four group. Strictly between the rationals and the whole field sit exactly three quadratic fields: the rationals with root 2, with root 3, and with root 6. Each one is fixed by a subgroup of order 2. Notice the reversal: the bigger the subgroup, the smaller the field it fixes.
The splitting field of x cubed minus 2 needs the real cube root plus the complex roots of unity. Adjoining the real root gives degree 3, and adjoining the unity roots doubles it to 6. So there are 6 automorphisms, forming the full symmetric group S3. Every subgroup of S3 matches one intermediate field, and inclusions run in opposite directions: a larger subgroup fixes a smaller field.
A group is solvable when it can be broken into abelian pieces. When the Galois group is solvable, the roots can be expressed with radicals. The symmetric group S5 has no such breakdown, so the general quintic admits no formula with radicals. To settle solvability questions, examine the group rather than the size of the roots.
Subgroups match intermediate fields in reverse order, and a solvable group means radical formulas exist.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.