Galois theory: Splitting fields

Summary

This lecture introduces the concept of splitting fields in Galois theory, focusing on finding the smallest extension field containing all roots of a given polynomial with coefficients in a base field. It explains the properties of splitting fields and their construction through examples, including linear, quadratic, and higher degree polynomials. The lecture also discusses the existence and uniqueness of splitting fields, highlighting potential ambiguities in their identification due to multiple isomorphisms.

Takeaways

• 🌟 A splitting field for a polynomial is an extension field that contains all the roots of the polynomial and where the polynomial factors into linear factors.
• 🔍 The goal is to find the smallest extension field that contains all the roots of a given polynomial with coefficients in the original field.
• 📐 For polynomials of degree one or two, finding a splitting field is straightforward as it involves adjoining a single root.
• 🔢 The concept of a splitting field is slightly imprecise because it involves not just the field itself but also an embedding of the original field into it.
• 📉 For higher degree polynomials, constructing a splitting field becomes more complex and may involve adjoining multiple roots or elements.
• 🔄 The process of finding a splitting field involves factoring the polynomial into irreducible components and adjoining roots step by step.
• 🔑 There is a uniqueness issue with splitting fields; different constructions can lead to isomorphic but non-identical fields.
• 🌐 The uniqueness of splitting fields up to isomorphism means that any two splitting fields of a polynomial over a field are isomorphic as extensions.
• 🔄 The degree of a splitting field over the original field can be calculated by considering the degrees of the field extensions involved in the construction.
• 🚀 Existence of splitting fields is guaranteed, but their uniqueness is only up to isomorphism, not in terms of specific elements or construction.

Q & A

What is a splitting field?

A splitting field is an extension field L of a field K such that a given polynomial p with coefficients in K factors into linear factors in L[x]. It is the smallest field extension containing all the roots of p.

Why is it important to find a splitting field?

Finding a splitting field is important because it allows us to understand the roots of a polynomial and their relationships within an extension of the original field. It also helps in simplifying the polynomial into linear factors, which is often useful in algebraic computations.

What are the two properties a field must have to be considered a splitting field?

A field L is a splitting field for a polynomial p over a field K if: 1) p factors into linear factors in L[x], and 2) L is the smallest field extension of K containing all the roots of p.

How is a splitting field constructed?

A splitting field is constructed by iteratively adjoining roots of irreducible polynomials to the original field until the polynomial factors completely into linear factors over the extension field.

What is the difference between a splitting field and a field extension that contains the roots of a polynomial?

A splitting field is the smallest field extension that contains all the roots of a polynomial and in which the polynomial factors into linear factors. A field extension that contains the roots may be larger and not necessarily factor the polynomial into linear factors.

Can you give an example of constructing a splitting field for a quadratic polynomial?

For a quadratic polynomial that is irreducible over a field K, a splitting field can be constructed by taking the ring of polynomials over K, quotienting by the ideal generated by the polynomial, resulting in a field that contains a root of the polynomial.

What happens when you construct a splitting field for a polynomial of degree greater than two?

For polynomials of degree greater than two, constructing a splitting field can become more complex. You may need to adjoin multiple roots and possibly deal with more intricate algebraic structures to ensure the polynomial factors into linear factors.

Are splitting fields always unique?

Splitting fields are not always unique in the sense that there can be multiple isomorphic splitting fields for a given polynomial. However, any two splitting fields of a polynomial over a field are isomorphic as extensions of that field.

What is the significance of the example involving the polynomial x^3 - 2 over the rational numbers?

The example of x^3 - 2 over the rational numbers illustrates that a field containing one root of a polynomial may not contain all roots. It shows the process of constructing a splitting field by adjoining roots step by step, resulting in a field extension of degree 6.

How does the construction of a splitting field relate to the concept of algebraic closures?

The construction of a splitting field is closely related to the concept of algebraic closures, as an algebraic closure of a field is the smallest algebraically closed field extension, which contains the original field and in which every non-constant polynomial splits into linear factors.

What is the practical application of splitting fields in the context of finite fields?

Splitting fields are useful in the construction of finite fields, which are important in various areas of mathematics and computer science, including error-correcting codes, cryptography, and finite geometry.