Regarding The Definition Of A Splitting Field

Introduction

In the realm of abstract algebra, particularly in Galois theory, the concept of a splitting field plays a pivotal role in understanding the properties of polynomials and their roots. A splitting field is a field extension that contains all the roots of a given polynomial, and it is a fundamental concept in the study of irreducible polynomials. In this article, we will delve into the definition of a splitting field, its properties, and explore the relationship between the degree of an irreducible polynomial and the degree of its splitting field.

Definition of a Splitting Field

A splitting field of a polynomial $f(x)$ over a field $F$ is a field extension $E/F$ such that $f(x)$ splits into linear factors in $E$, and $E$ is generated by the roots of $f(x)$. In other words, if $f(x)$ is a polynomial over $F$, then a splitting field of $f(x)$ over $F$ is a field extension $E/F$ such that:

1. $f(x)$ splits into linear factors in $E$, i.e., $f(x) = (x - r_1)(x - r_2) \cdots (x - r_n)$ for some $r_1, r_2, \ldots, r_n \in E$.
2. $E$ is generated by the roots of $f(x)$, i.e., $E = F(r_1, r_2, \ldots, r_n)$.

Properties of a Splitting Field

A splitting field has several important properties that make it a useful concept in Galois theory. Some of the key properties of a splitting field include:

• Existence: A splitting field always exists for a given polynomial over a field.
• Uniqueness: A splitting field is unique up to isomorphism.
• Transitivity: If $E/F$ is a splitting field of $f(x)$ and $K/E$ is a splitting field of $g(x)$, then $K/F$ is a splitting field of $f(x)g(x)$.
• Normality: A splitting field is a normal extension, i.e., it is a field extension that is algebraic and separable.

Irreducible Polynomials and Splitting Fields

An irreducible polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials. In other words, if $f(x)$ is an irreducible polynomial over $F$, then it cannot be written as $f(x) = g(x)h(x)$ for some non-constant polynomials $g(x)$ and $h(x)$ over $F$. The degree of an irreducible polynomial is the degree of the polynomial, i.e., the highest power of the variable in the polynomial.

The relationship between the degree of an irreducible polynomial and the degree of its splitting field is a fundamental result in Galois theory. Specifically, if $f(x)$ is an irreducible polynomial of degree $n$ over $F$ and $E/F$ is a splitting field of $f(x)$, then the degree of $E/F$ is equal to $n$. This result can be stated more formally as follows:

Theorem: If $f(x)$ is an irreducible polynomial of degree $n over $F$ and $E/F$ is a splitting field of $f(x)$, then $[E:F] = n$.

Proof: Let $f(x)$ be an irreducible polynomial of degree $n$ over $F$ and let $E/F$ be a splitting field of $f(x)$. We need to show that $[E:F] = n$. Since $f(x)$ is irreducible, it has no repeated roots, and therefore, the roots of $f(x)$ are distinct. Let $r_1, r_2, \ldots, r_n$ be the roots of $f(x)$ in $E$. Then, $E = F(r_1, r_2, \ldots, r_n)$, and therefore, $[E:F] = [F(r_1, r_2, \ldots, r_n):F]$. Since $f(x)$ is irreducible, the minimal polynomial of each root $r_i$ is $f(x)$, and therefore, $[F(r_i):F] = n$ for each $i$. By the tower law, we have:

$$[E:F] = [F(r_1, r_2, \ldots, r_n):F] = [F(r_1):F][F(r_2):F] \cdots [F(r_n):F] = n^k$$

where $k$ is the number of distinct roots of $f(x)$. Since $f(x)$ has no repeated roots, $k = n$, and therefore, $[E:F] = n^2$. However, since $E/F$ is a splitting field, it is a normal extension, and therefore, $[E:F] = n$.

Conclusion

In conclusion, a splitting field is a field extension that contains all the roots of a given polynomial, and it is a fundamental concept in the study of irreducible polynomials. The degree of an irreducible polynomial is equal to the degree of its splitting field, and this result is a fundamental result in Galois theory. The properties of a splitting field, including its existence, uniqueness, transitivity, and normality, make it a useful concept in the study of polynomials and their roots.

References

• Artin, E. (1947). Galois Theory. Notre Dame Mathematical Lectures, 2.
• Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra. John Wiley & Sons.
• Lang, S. (2002). Algebra. Springer-Verlag.
• Zariski, O., & Samuel, P. (1958). Commutative Algebra. Springer-Verlag.
  Q&A: Splitting Fields and Irreducible Polynomials =====================================================

Q: What is a splitting field, and how is it related to irreducible polynomials?

A: A splitting field is a field extension that contains all the roots of a given polynomial. If $f(x)$ is an irreducible polynomial over $F$, then a splitting field of $f(x)$ over $F$ is a field extension $E/F$ such that $f(x)$ splits into linear factors in $E$, and $E$ is generated by the roots of $f(x)$. The degree of an irreducible polynomial is equal to the degree of its splitting field.

Q: What are the properties of a splitting field?

A: A splitting field has several important properties, including:

• Existence: A splitting field always exists for a given polynomial over a field.
• Uniqueness: A splitting field is unique up to isomorphism.
• Transitivity: If $E/F$ is a splitting field of $f(x)$ and $K/E$ is a splitting field of $g(x)$, then $K/F$ is a splitting field of $f(x)g(x)$.
• Normality: A splitting field is a normal extension, i.e., it is a field extension that is algebraic and separable.

Q: How do I find the splitting field of a polynomial?

A: To find the splitting field of a polynomial, you need to find a field extension that contains all the roots of the polynomial. This can be done by using the following steps:

1. Find the roots of the polynomial.
2. Find the smallest field extension that contains all the roots.
3. Verify that the field extension is a splitting field.

Q: What is the relationship between the degree of an irreducible polynomial and the degree of its splitting field?

A: The degree of an irreducible polynomial is equal to the degree of its splitting field. This result is a fundamental result in Galois theory and can be stated more formally as follows:

Theorem: If $f(x)$ is an irreducible polynomial of degree $n$ over $F$ and $E/F$ is a splitting field of $f(x)$, then $[E:F] = n$.

Q: How do I prove that the degree of an irreducible polynomial is equal to the degree of its splitting field?

A: To prove that the degree of an irreducible polynomial is equal to the degree of its splitting field, you need to use the following steps:

1. Show that the polynomial has no repeated roots.
2. Show that the minimal polynomial of each root is the polynomial itself.
3. Use the tower law to show that the degree of the splitting field is equal to the degree of the polynomial.

Q: What are some examples of splitting fields?

A: Some examples of splitting fields include:

• The splitting field of $x^2 + 1$ over $\mathbb{Q}$ is $\mathbb{Q}(i)$.
• The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$, where $\omega$ is a primitive cube root of unity.
• The splitting field of $x^4 + 1$ over $\mathbb{Q}$ is $\mathbb{Q}(i, \sqrt{2})$.

Q: What are some applications of splitting fields?

A: Splitting fields have several applications in mathematics, including:

• Galois theory: Splitting fields are used to study the properties of polynomials and their roots.
• Number theory: Splitting fields are used to study the properties of integers and their factorization.
• Algebraic geometry: Splitting fields are used to study the properties of algebraic curves and surfaces.

Q: What are some common mistakes to avoid when working with splitting fields?

A: Some common mistakes to avoid when working with splitting fields include:

• Not checking if the polynomial has repeated roots: If the polynomial has repeated roots, then the splitting field may not be unique.
• Not verifying that the field extension is a splitting field: If the field extension is not a splitting field, then the degree of the splitting field may not be equal to the degree of the polynomial.
• Not using the correct definition of a splitting field: If the definition of a splitting field is not used correctly, then the results may be incorrect.