Abstract Algebra

Field

In abstract algebra, a field is a set equipped with two operations, usually called addition and multiplication. These operations satisfy several properties, which we outline below.

1. Closure: For any two elements \( a \) and \( b \) in a field \( F \), the sum \( a + b \) and the product \( a \cdot b \) are also in \( F \).
2. Associativity: Addition and multiplication are associative, meaning that \( (a+b)+c = a+(b+c) \) and \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for all \( a, b, \) and \( c \) in \( F \).
3. Commutativity: Addition and multiplication are commutative, meaning that \( a + b = b + a \) and \( a \cdot b = b \cdot a \) for all \( a \) and \( b \) in \( F \).
4. Existence of Identity Elements: There exist two distinct identity elements: an additive identity \( 0 \) and a multiplicative identity \( 1 \), such that for all \( a \) in \( F \), \( a + 0 = a \) and \( a \cdot 1 = a \).
5. Existence of Inverses: Every element \( a \) in \( F \) (except for the additive identity \( 0 \)) has an additive inverse \( -a \) such that \( a + (-a) = 0 \). Additionally, every element \( a \) in \( F \) (except for the multiplicative identity \( 1 \) and the additive identity \( 0 \)) has a multiplicative inverse \( a^{-1} \) such that \( a \cdot a^{-1} = 1 \).
6. Distributivity: Multiplication distributes over addition, meaning that \( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \) for all \( a, b, \) and \( c \) in \( F \).

Group

A group \( G \) is a set equipped with a binary operation \( \cdot \), denoted as \( (G, \cdot) \), satisfying the following properties:

1. Closure: For any two elements \( a, b \) in \( G \), the result of \( a \cdot b \) is also in \( G \).

2. Associativity: The operation \( \cdot \) is associative, meaning that \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for all \( a, b, c \) in \( G \).

3. Identity Element: There exists an element \( e \) in \( G \) such that for any element \( a \) in \( G \), \( a \cdot e = e \cdot a = a \). This element \( e \) is called the identity element of \( G \).

4. Inverse Element: For every element \( a \) in \( G \), there exists an element \( b \) in \( G \) such that \( a \cdot b = b \cdot a = e \), where \( e \) is the identity element of \( G \). The element \( b \) is called the inverse of \( a \) and is denoted as \( a^{-1} \).

Distinguish between field and group

1. Definition:
   • Field: A field is a set equipped with two operations, typically addition and multiplication, that satisfy specific properties. These properties include closure, associativity, commutativity, distributivity, existence of additive and multiplicative identities, and existence of additive inverses and multiplicative inverses (except for the additive identity).
   • Group: A group is a set equipped with a single binary operation that satisfies closure, associativity, existence of an identity element, and existence of inverses for every element.
2. Operations:
   • Field: Fields have two operations: addition and multiplication. These operations satisfy the properties mentioned above.
   • Group: Groups have a single operation, typically denoted as *, which combines two elements of the group to produce another element of the group. This operation also satisfies the group properties mentioned above.
3. Inverses:
   • Field: In a field, every nonzero element has a multiplicative inverse. Additionally, every element has an additive inverse.
   • Group: In a group, every element has an inverse with respect to the group operation.
4. Example:
   • Field: The set of real numbers \(R\) forms a field under addition and multiplication. Every real number except 0 has a multiplicative inverse.
   • Group: The set of integers \(Z\) forms a group under addition. The identity element is 0, and every integer has an additive inverse.

Polynomial ring

 

A polynomial ring is a ring that consists of polynomials with coefficients from a given ring (often a field) and with one or more variables.

For example:

• Polynomial ring in one variable \( R[x] \):
  • Here, \( R \) is a ring, and \( x \) is an indeterminate (a variable).
  • \( R[x] \) represents all possible polynomials where the coefficients are elements of \( R \).
  • An example would be \( \mathbb{R}[x] \), which is the ring of all polynomials with real number coefficients.
  • Elements of \( \mathbb{R}[x] \) look like this: \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_i \in \mathbb{R} \) (the coefficients) and \( n \geq 0 \).
• Polynomial ring in multiple variables \( R[x_1, x_2, \ldots, x_n] \):
  • If there are multiple variables, you get polynomials in more than one variable.
  • For example, \( \mathbb{R}[x, y] \) is the ring of polynomials with coefficients in \( \mathbb{R} \) and variables \( x \) and \( y \).
  • An element of \( \mathbb{R}[x, y] \) might look like \( 3x^2y + 2xy – 5y^3 + 7 \).

Key Properties of Polynomial Rings

1. Coefficients from a Ring:
   • The coefficients of the polynomials come from a base ring \( R \), which is often a field like \( \mathbb{R} \) (real numbers), \( \mathbb{Z} \) (integers), or \( F_p \) (a finite field with \( p \) elements).
   • \( R \) must have a well-defined addition, subtraction, and multiplication to ensure the polynomial ring has these properties too.
2. Operations:
   • Addition: Polynomials are added by adding their coefficients for like terms.

     Example: \( (3x^2 + 2x + 1) + (x^2 – x + 4) = 4x^2 + x + 5 \).

   • Multiplication: Polynomials are multiplied using the distributive law.

     Example: \( (x + 2)(x – 3) = x^2 – 3x + 2x – 6 = x^2 – x – 6 \).

3. Non-Commutativity with Variables:
   • The ring \( R \) might be commutative (where \( a \cdot b = b \cdot a \) for all \( a, b \in R \)), but in a polynomial ring, when there are multiple variables, the multiplication order can matter.
   • For example, in \( \mathbb{R}[x, y] \), \( xy \) is generally considered the same as \( yx \), but if you introduce relations or constraints (like in a quotient ring), this may change.

Example: Finite Fields and Polynomial Rings

A common example in cryptography is the polynomial ring over a finite field \( F_p \), such as \( F_p[x] \):

• \( F_p[x] \) represents the set of all polynomials with coefficients in the finite field \( F_p \), where \( p \) is a prime number.
• The coefficients of each polynomial are integers modulo \( p \).

Summary

A polynomial ring \( R[x] \) or \( R[x_1, x_2, \ldots, x_n] \):

• Is a set of polynomials with coefficients in a ring \( R \).
• Supports polynomial addition, subtraction, and multiplication.
• Provides a framework to study algebraic structures and solve equations.

Understanding polynomial rings is fundamental for more advanced concepts like quotient rings and ideals, where polynomials are constrained by certain relations, such as in \( F_p[x, y]/(x^2 + y^2 – 1) \), which introduces constraints based on an algebraic curve.

multivariate polynomial rings

For multivariate polynomial rings, there are generally no inherent restrictions on the degrees of the variables unless specified by the context in which the polynomial ring is being used. Here’s a breakdown of the concept and typical considerations regarding the degrees of variables in a multivariate polynomial ring:

Degrees in Multivariate Polynomial Rings

1. Multivariate Polynomial Ring Structure:
   • A multivariate polynomial ring \( R[x_1, x_2, \ldots, x_n] \) consists of polynomials with coefficients from a ring \( R \) (often a field), and variables \( x_1, x_2, \ldots, x_n \).
   • An element of \( R[x_1, x_2, \ldots, x_n] \) might look like this:

     \[ f(x_1, x_2, \ldots, x_n) = a_1 x_1^2 x_2 + a_2 x_2^3 + a_3 x_1 x_2^2 x_3 + \ldots \]
     where \( a_1, a_2, a_3, \ldots \in R \).

2. Total Degree:
   • The total degree of a multivariate polynomial is the highest sum of exponents of all variables in any single term.
   • For example, if \( f(x, y) = 3x^2y + 4y^3 + 2xy^2 \), then:
     • The degree of the term \( 3x^2y \) is \( 2 + 1 = 3 \).
     • The degree of \( 4y^3 \) is \( 3 \).
     • The degree of \( 2xy^2 \) is \( 1 + 2 = 3 \).
     • Thus, the total degree of \( f \) is \( 3 \).
3. Partial Degrees:
   • Each variable can have its own partial degree, which is the highest exponent of that specific variable in the polynomial.
   • For example, in \( f(x, y, z) = 2x^3y^2 + 3yz + 5x \):
     • The degree in \( x \) is \( 3 \) (from \( 2x^3y^2 \)).
     • The degree in \( y \) is \( 2 \) (from \( 2x^3y^2 \)).
     • The degree in \( z \) is \( 1 \) (from \( 3yz \)).
4. No Intrinsic Constraints on Degrees:
   • By default, in the definition of a polynomial ring \( R[x_1, x_2, \ldots, x_n] \), there is no limit on the degree of each variable.
   • You can have terms like \( x_1^{10} x_2^5 \) or \( x_1 x_2^{100} \). The degree of each variable is only limited by the terms present in a given polynomial.
   • However, in some specific applications or contexts, bounded-degree polynomial rings are used, where the degrees of the variables are constrained. This is often done to simplify computations or because the structure being modeled (e.g., an algebraic curve) imposes such restrictions.
5. Degree Constraints in Specific Applications:
   • Coding Theory: When using polynomials for error-correcting codes, constraints might be applied to ensure that polynomials are of a certain degree or less.
   • Cryptography: In cryptographic protocols, such as those involving finite fields or algebraic geometry codes, polynomial degrees might be limited to ensure efficient computation.
   • Algebraic Geometry: When defining varieties or solving systems of polynomial equations, it’s common to work with polynomials of a specific degree to ensure that solutions remain within certain bounds.

Summary

• In a multivariate polynomial ring \( R[x_1, x_2, \ldots, x_n] \), polynomials can have terms of any degree in each variable.
• The total degree is the sum of exponents in the highest-degree term, while partial degrees refer to the highest exponent of each individual variable.
• There are no default restrictions on the degree of the variables in a multivariate polynomial ring, but specific applications might impose such constraints for practical or theoretical reasons.

A ring is an algebraic structure that consists of a set equipped with two operations: addition (+) and multiplication (*). These operations must satisfy specific properties like associativity, distributivity, and the existence of an additive identity (0). Rings are fundamental in abstract algebra, providing a framework for working with many familiar number systems, such as integers.

Properties of a Ring

• Addition:
  • Forms an abelian group, meaning that addition is associative, has an identity element (0), and every element has an additive inverse.
• Multiplication:
  • Associativity: The multiplication operation must be associative, meaning \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \) for any elements \( a \), \( b \), and \( c \) in the ring.
  • Distributivity: Multiplication must distribute over addition, meaning \( a \cdot (b + c) = a \cdot b + a \cdot c \) and \( (a + b) \cdot c = a \cdot c + b \cdot c \).
• Optional Properties:
  • A ring may or may not have a multiplicative identity (1) such that \( a \cdot 1 = a \) for all \( a \) in the ring.
  • Multiplication does not have to be commutative; that is, it is not required that \( a \cdot b = b \cdot a \) for all \( a \) and \( b \).

Example: Ring of Integers

The set of integers \( \mathbb{Z} \) with standard addition and multiplication is a common example of a ring:

• Addition: Forms an abelian group, with 0 as the additive identity and \( -a \) as the additive inverse of \( a \).
• Multiplication: Is associative and distributes over addition. However, not every integer has a multiplicative inverse (e.g., there is no integer \( x \) such that \( 2 \cdot x = 1 \)).
• Multiplicative Identity: The multiplicative identity is 1, as \( a \cdot 1 = a \) for any integer \( a \).

Summary

In summary, a ring:

• Extends the structure of a group with a second operation (multiplication).
• Must satisfy associativity and distributivity properties for its operations.
• May or may not have a multiplicative identity or require multiplication to be commutative.

Rings provide the foundation for more complex algebraic structures, such as fields and polynomial rings.

Property	Group	Ring	Field
Operations	1 (e.g., addition or multiplication)	2 (addition and multiplication)	2 (addition and multiplication)
Associativity	Required for the single operation	Required for both operations	Required for both operations
Commutativity	Not required (unless Abelian)	Not required (often required for addition)	Required for both addition and multiplication
Identity	Required (for addition or multiplication)	Required (additive identity, optional multiplicative)	Required (additive and multiplicative identities)
Inverses	Required for all elements (additive or multiplicative)	Additive inverses required; multiplicative inverses not required	Required for both addition and multiplication (non-zero elements)
Distributivity	Not applicable	Required	Required
Examples	ℤ, + (integers under addition)	ℤ (integers)	ℚ (rationals), ℝ (reals), 𝔽p (finite fields)