The smallest field containing all roots of a polynomial -- where symmetry lives
Some polynomials only fall apart completely once you add enough roots. The smallest field where that happens is the splitting field, and it is where Galois theory lives.
Pick a polynomial and press Adjoin First Root, then keep adjoining. Notice that some roots come for free: adjoining one can force others into the field.
Polynomial: x^2 - 2
The splitting field of a polynomial f(x) over F is the smallest extension of F that contains all the roots of f. It is where Galois theory lives: the splitting field is always a normal extension, meaning that every irreducible polynomial that has one root in the field has all its roots there.
Pick a polynomial and watch it factor one root at a time. As each root is adjoined, the field tower grows and the polynomial factorization updates. Notice how some roots "come for free" -- adjoining one root may force others into the field automatically.
Key insight: For x^2 - 2, adjoining sqrt(2) gives -sqrt(2) for free (it equals -1 times sqrt(2)). For x^3 - 2, adjoining the real cube root does NOT give the complex roots for free -- you need to also adjoin a primitive cube root of unity omega. This is why the splitting field of x^3 - 2 has degree 6, not 3.
Switch between examples and compare filled circles (roots in the field) with hollow ones (conjugates that are missing). A normal extension has no hollow circles.
Minimal polynomial: x^2 - 2
Q(sqrt(2))
2 / 2 roots present in field
Both roots of x^2 - 2 lie in Q(sqrt(2)). Since all roots of the minimal polynomial are present, this is a normal extension.
An extension E/F is normal if every irreducible polynomial over F that has one root in E splits completely in E. Equivalently, E contains all conjugates of each of its elements. Splitting fields are always normal; arbitrary extensions need not be.
Key insight: Q(sqrt(2))/Q is normal because x^2 - 2 has both roots (+/-sqrt(2)) in Q(sqrt(2)). But Q(cbrt(2))/Q is NOT normal: x^3 - 2 has only one root (the real cube root) in Q(cbrt(2)), while the other two roots omega*cbrt(2) and omega^2*cbrt(2) are complex and missing.
Galois extensions = normal + separable -- the splitting field of a separable polynomial is always Galois
Select a polynomial and press Begin Splitting to factor it step by step. The end screen shows the splitting field, its degree, a basis, and the roots with lines linking conjugates.
Degree 2 polynomial -- select to compute its splitting field step by step
Select a polynomial and compute its splitting field step by step. The factorization tree shows how the polynomial breaks apart over successive extensions. The final result displays the splitting field, its degree over Q, a basis, and all roots with symmetry lines connecting conjugates.
Key insight: The degree of the splitting field divides n! (where n is the degree of the polynomial), but is often much smaller. For x^4 - 5x^2 + 6 = (x^2-2)(x^2-3), the splitting field Q(sqrt(2), sqrt(3)) has degree 4, not 4! = 24, because the roots are related by simple algebraic identities.
Splitting field -- the smallest field containing all roots of a polynomial; unique up to isomorphism
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