Problem 1

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.) Show that, in any extension of $\mathbb{Z}_{3}$ which contains a root $u$ of $$ a(x)=x^{3}+2 x+1 \in \mathbb{Z}_{3}[x] $$ it happens that $u+1$ and $u+2$ are the remaining two roots of $a(x)$. Use this fact to find the root field of $x^{3}+2 x+1$ over $\mathbb{Z}_{3}$. Write its addition and multiplication tables

Step-by-Step Solution

Verified

Answer

The root field is $ \mathbb{Z}_3(u) $, containing elements $ u, u+1, u+2 $, with addition and multiplication tables constructed from these elements.

1Step 1: Understand the polynomial in question

The given polynomial is $ a(x) = x^3 + 2x + 1 $ over $ \mathbb{Z}_3 $, a field with three elements: 0, 1, and 2.

2Step 2: Identify a root of the polynomial

Substitute $ x = 0, 1, 2 $ into $ a(x) = x^3 + 2x + 1 $ and check if any of these values satisfy the equation under modulo 3 arithmetic. After testing, you will find that $ x = 1 $ is a root because $1^3 + 2\cdot1 + 1 \equiv 0 \pmod{3}$.

3Step 3: Use properties of roots in extensions of $ \mathbb{Z}_3 $

If $ u $ is a root of $ a(x) $, it is known that $ u+1 $ and $ u+2 $ also need to be roots of $ a(x) $. This can be seen by plugging $ u+1 $ and $ u+2 $ back into $ a(x) $ and showing $ a(u) = a(u+1) = a(u+2) = 0 $.

4Step 4: Determine the root field of the polynomial

Since the polynomial $ x^3 + 2x + 1 $ is of degree 3 and we found three roots $ u, u+1, u+2 $, it's clear this polynomial splits in its root field. Thus, the root field is the smallest field containing $ \mathbb{Z}_3 $ and the roots, denoted as $ \mathbb{Z}_3(u) $.

5Step 5: Construct the addition and multiplication tables for the root field

The root field, $ \mathbb{Z}_3(u) $, will have 27 elements because it is an extension of degree 3 of $ \mathbb{Z}_3 $. Define the elements and construct the addition and multiplication tables by simplifying expressions under modulo 3 arithmetic.

Key Concepts

Field ExtensionsRoot FieldsPolynomials Over Finite FieldsGalois Theory

Field Extensions

In algebra, a field extension represents a larger field that includes a smaller base field and additional elements which satisfy certain polynomial equations. For example, when considering a finite field like $ \mathbb{Z}_3 $, if we have a polynomial like $ x^2 + 1 $, it may not have roots in $ \mathbb{Z}_3 $ itself. Thus, we extend $ \mathbb{Z}_3 $ to a bigger field $ \mathbb{Z}_3(u) $, where $ u $ is a root of the polynomial. This means $ u^2 + 1 = 0 $ holds true within this new field.

To facilitate working with this new field, both addition and multiplication operations are defined. The elements you can generate through combinations of $ \mathbb{Z}_3 $ and $ u $ are the entirety of this field. This is vital when solving equations that initially seem unsolvable within a base smaller field.

Field extensions allow us to find solutions to polynomial equations not solvable within the original field.
The degree of the field extension corresponds to the number of new elements or roots introduced.
Understanding these concepts is foundational for deeper topics like Galois Theory.

Root Fields

A root field is a specific type of field extension where all roots of a polynomial are included. Consider the polynomial $ x^3 + 2x + 1 $ over $ \mathbb{Z}_3 $. It might initially appear difficult to solve, because its roots aren't visible within $ \mathbb{Z}_3 $. However, by determining one root and using properties of polynomial equations, the entire polynomial can be expressed in its root field.

Upon trial, we find the root $ u = 1 $ of this polynomial satisfies the condition in $ \mathbb{Z}_3 $. By applying known properties of roots and polynomial roots in extensions, it follows that $ u+1 $ and $ u+2 $, are also roots. All these roots together form the root field $ \mathbb{Z}_3(u) $. This is a powerful tool often necessitated in algebraic geometry and number theory.

Root fields provide the smallest field in which a polynomial splits completely into linear factors.
This powerful tool aids in understanding the structure of solutions to polynomial equations.

Polynomials Over Finite Fields

Working with polynomials over finite fields involves special arithmetic rules. For example, a finite field like $ \mathbb{Z}_3 $ contains only the elements 0, 1, and 2. Arithmetic here uses modulo operations – any operation result is wrapped back within these values. This behavior poses unique challenges when finding polynomial roots, as polynomials may not factor easily.

In simpler terms, leading coefficients, solutions, and degree considerations must always respect these mod operations. For instance, when testing potential roots of $ x^3 + 2x + 1 $, substitutes like 0, 1, and 2 must be put through the polynomial with results evaluated over mod 3. Understanding how such polynomials operate is crucial in several fields, particularly cryptography.

Finite fields, often used in encryption, operate within a closed arithmetic environment.
Unique behaviors emerge from simple mod arithmetic, challenging usual polynomial factorization methods.

Galois Theory

Galois Theory is an advanced mathematical concept that connects field theory and group theory. It studies the symmetries of polynomial equations, providing a bridge between algebra and geometry. In our case, with polynomials over fields like $ \mathbb{Z}_3 $, it helps explore why a polynomial has certain roots and what structural properties their field extensions possess.

When dealing with a polynomial such as $ x^3 + 2x + 1 $, Galois Theory offers insight into why the roots $ u, u+1, $ and $ u+2 $ exist and are structurally significant. Each extension corresponds to a group that symmetrically represents these extensions. By applying Galois Theory, one can predict and verify relationships between polynomial roots fundamentally, even diving into unsolvable equations over simple fields.

Galois Theory provides a comprehensive method to understand why and how certain polynomials extend fields.
The interrelations illuminated by Galois spirits simplify complex algebraic problems, emphasizing symmetry and transformations.

Problem 1

Problem 2

Other exercises in this chapter

Problem 1

Let $F$ be a field. An irreducible polynomial $p(x)$ in $F[x]$ is said to be separable over $F$ if it has no multiple roots in any extension of $F$. I

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Problem 1

Find $c$ such that $Q(\sqrt{2}, \sqrt{-3})=Q(c)$. Do the same for $\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})$.

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Problem 2

If $\mathbb{Z}_{p}[y]$ is the domain of polynomials (in the letter $y$ ) over $\mathbb{Z}_{p}$, let $E=\mathbb{Z}_{p}(y)$ be the field of quotients of \

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Problem 2

If $F \subseteq I \subseteq K$ and $K$ is a root field of $a(x)$ over $F$, then $K$ is a root field of $a(x)$ over $I .$

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