Problem 2 - A Book of Abstract Algebra

Question

Find the root field of \(\mathrm{x}^{2}+1\) over \(\mathbb{Z}_{3} .\) ANSWER By the basic theorem of field extensions, $$ \mathbb{Z}_{3}[x] /\left\langle x^{2}+1\right\rangle \cong \mathbb{Z}_{3}(u) $$ where \(u\) is a root of \(x^{2}+1 .\) In \(\mathbb{Z}_{3}(u), x^{2}+1=(x+u)(x-u)\), because \(u^{2}+1=0\). Since \(\mathbb{Z}_{3}(u)\) contains \(\pm u\), it is the root field of \(x^{2}+1\) over \(\mathbb{Z}_{3} .\) Note that \(\mathbb{Z}_{3}(u)\) has nine elements, and its addition and multiplication tables are easy to construct. (See Chapter 27, Exercise C.) 2 Find the root field of \(x^{3}+x^{2}+x+2\) over \(\mathbb{Z}_{3}\), and write its addition and multiplication tables.

Step-by-Step Solution

Answer

The root field is \(\mathbb{Z}_3(\alpha)\), where \(\alpha^3 + \alpha^2 + \alpha + 2 = 0\).

Step 1: Understanding the Problem

We need to find the root field of the polynomial \(x^3 + x^2 + x + 2\) over the finite field \(\mathbb{Z}_3\). A root field here refers to a field extension of \(\mathbb{Z}_3\) in which the polynomial factors completely and each of its roots belong to the field.

Step 2: Reducing Polynomial in \(\mathbb{Z}_3[x]\)

Since we are over \(\mathbb{Z}_3\), our polynomial simplifies with coefficients as is: \(x^3 + x^2 + x + 2\). The coefficients are already reduced modulo 3.

Step 3: Checking Irreducibility

We should check if \(x^3 + x^2 + x + 2\) is irreducible over \(\mathbb{Z}_3\). A cubic polynomial is reducible if it has a root in \(\mathbb{Z}_3\). By testing \(x = 0, 1, 2\), we find no roots since replacing \(x\) in the polynomial does not yield zero.

Step 4: Constructing the Root Field

Since \(x^3 + x^2 + x + 2\) is irreducible, the root field is an extension of degree 3. Let \(\alpha\) be a root of the polynomial. The root field can be described as \(\mathbb{Z}_3(\alpha)\), where \(\alpha^3 + \alpha^2 + \alpha + 2 = 0\).

Step 5: Determining Field Structure

\(\mathbb{Z}_3(\alpha)\) will consist of elements \{0, 1, 2, \alpha, \alpha + 1, \alpha + 2, 2\alpha, 2\alpha + 1, 2\alpha + 2\}. With this, the addition and multiplication tables can be created similarly to modules over \(\mathbb{Z}_3\) using properties of \(\alpha\).

Key Concepts

Finite Fields

Finite fields, also known as Galois fields, are fields with a finite number of elements. Each finite field \( \mathbb{F}_q \) consists of \( q \) elements, where \( q \) is a power of a prime number. These fields are pivotal in various applications, including coding theory, cryptography, and digital signaling. Because finite fields have a usefully limited set of elements, their arithmetic operations can be performed efficiently.
Understanding finite fields begins with considering the simple prime fields like \( \mathbb{Z}_3 \). In this example, elements of \( \mathbb{Z}_3 \) are \{0, 1, 2\}, and addition or multiplication of these elements is done modulo 3. This essentially means that we "wrap around" upon reaching the prime number 3, allowing us to effectively work within a bounded system.

• For addition: \( 2 + 1 \equiv 0 \pmod{3} \)
• For multiplication: \( 2 \cdot 2 \equiv 1 \pmod{3} \)

Recognizing the simplicity of operations in finite fields helps us appreciate their value in constructing more complex field extensions.

Polynomial Irreducibility

Polynomial irreducibility is a concept that helps identify whether a polynomial can be simplified into polynomials of lower degrees over a certain field. A polynomial is considered irreducible over a field if it cannot be factored further, apart from multiplying by units.
For example, with the polynomial \( x^3 + x^2 + x + 2 \) over \( \mathbb{Z}_3 \), checking for irreducibility involves testing if this polynomial has any roots in \( \mathbb{Z}_3 \). A cubic polynomial like this is irreducible if substituting \( x \) with any element in the field (in this case, 0, 1, 2) does not yield zero. This lack of roots confirms that it cannot be factored further into lower degree polynomials within \( \mathbb{Z}_3 \).
Understanding irreducibility is crucial as it indicates that the polynomial can form a basis for a field extension, leading to the construction of a larger yet structured mathematical space.

Field Extensions

Field extensions create larger fields from smaller ones by adding new elements that fulfill specific equations. They are instrumental in solving polynomials that can't be solved within the initial field. The degree of a field extension indicates how many times larger the new field is compared to the original.
In the context of the given polynomial \( x^3 + x^2 + x + 2 \), an irreducible polynomial of degree 3 over \( \mathbb{Z}_3 \), the field extension knows as \( \mathbb{Z}_3(\alpha) \) is required, where \( \alpha \) is a root of the polynomial. This extension is a result of no roots within the original field and takes three dimensions.

• The original field, \( \mathbb{Z}_3 \), consists of 3 elements.
• The extension \( \mathbb{Z}_3(\alpha) \) includes elements that form a 3-degree vector space over \( \mathbb{Z}_3 \).

Each new element in the field extension helps solve equations that were deemed insolvable in the smaller field.

Root Fields

Root fields, also known as splitting fields, are essential in algebra, offering comprehensive fields where polynomials fully factor into linear factors. These fields contain all the roots of a polynomial, resulting in the polynomial's complete factorization.
When we discuss a root field for a polynomial like \( x^2 + 1 \) over \( \mathbb{Z}_3 \), it is a field where this polynomial’s forces complete factorization, incorporating all essential roots. Such a construction highlights the algebraic closure properties in practice.
The root field, typically denoted as \( \mathbb{Z}_3(u) \) for this polynomial, where \( u \) is the principal root, allows us a deeper insight into the polynomial's structure. Although it represents an augmented field over \( \mathbb{Z}_3 \), it enables us to derive all the necessary roots and view the field algebraically enclosed with respect to the polynomial under consideration.
Understanding root fields involves not just solving individual equations, but also grasping how these solutions integrate into broader field structures, offering richer algebraic contexts.