# Are Splitting Fields Unique for Different Polynomial Families?

• I
• PsychonautQQ
In summary, if both E and E' are extensions of K that are splitting fields for different families of polynomials in K[x], they may not be isomorphic. This is because they could have the same zeros, but the sets may be different. To determine isomorphism, one must consider the ideal generated by the two families.
PsychonautQQ
So if E and E' are both extensions of K so that both E and E' are splitting fields of different families of polynomials in K[x], then E and E' are not isomorphic, correct? They need to be splitting fields for the same family of polynomials in K[x], correct?

PsychonautQQ said:
So if E and E' are both extensions of K so that both E and E' are splitting fields of different families of polynomials in K[x], then E and E' are not isomorphic, correct? They need to be splitting fields for the same family of polynomials in K[x], correct?
I don't think so, at least not in this generality. They could still have the same zeros although the sets might be different. I think one has to consider the ideal generated by the two families.

## 1. What is a splitting field?

A splitting field is a field extension of a given field, in which a given polynomial completely factors into linear factors. In other words, all the roots of the polynomial are contained in the splitting field.

## 2. Why is the uniqueness of splitting fields important?

The uniqueness of splitting fields is important because it allows us to define a unique field extension for a given polynomial, which can then be used to study the properties and characteristics of that polynomial.

## 3. How is the uniqueness of splitting fields proven?

The uniqueness of splitting fields can be proven using the fundamental theorem of Galois theory. This theorem states that for a given polynomial, there is a one-to-one correspondence between its roots and the automorphisms of its splitting field. Therefore, any two splitting fields for the same polynomial must be isomorphic.

## 4. Can a polynomial have multiple splitting fields?

No, a polynomial can only have one splitting field. This is because the splitting field is uniquely determined by the polynomial, and any other field extension that also satisfies the definition of a splitting field would be isomorphic to the original one.

## 5. How are splitting fields related to algebraic closure?

Splitting fields are a crucial step in constructing the algebraic closure of a field. The algebraic closure of a field is the smallest field extension in which all polynomials have roots. This is achieved by repeatedly taking splitting fields until all polynomials have completely factored into linear factors.

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