Algebraic Number Theory is a historical mathematical concept that still has relevant bearing in today’s mathematics. Majorly all types of calculations are based on this theory, and as such, it is very important that students understand it. The mathematical concepts and definitions of integers and rational and irrational numbers are given meaning by the algebraic number theory.
It is important to understand that the Algebraic Number theorem does not focus so much on calculations and is more about explanations based on the number theorem. This field of mathematics is quite large, and as such, students will need to pay attention to this basic introduction to understand all that is being done and how they can apply them when necessary.
This piece will explain as easily as possible every definition so that students can understand Algebra Number Theory right from its basics and why it is so much applied in recent times.
What is Algebraic Number Theory?
Algebraic Number Theory is a branch of mathematics that is under the Number Theory. This theory uses Abstract Algebra techniques to study and make conclusions on rational numbers, Integers and their Generalizations. Abstract Number Theory is a vast branch of number theory that drives meaning to virtually every concept and study of mathematics. Understanding the entire concept of Algebraic Number Theory requires sticking with some of the most important definitions like elements, fields and rings. The following sections will consider them step by step.
The notation for complex numbers is C
The notation for Real numbers is R
The notation for Rational numbers is Q
What are Elements?
In mathematics and Algebraic theory, an Element is a distinct object that is considered a member of a set. For example;
A = [3, 6, 9, 12] is an example of a set and elements. In this case, A is the set. While 3, 5, 9 and 12 are all distinct elements of set A. The type of elements, in this case, are real numbers.
What is a Field?
A field denoted by F is a set with two major binary operations given as + and X. However, this definition is just classic. In the real sense, a field is any set where the operations, addition, multiplication, subtraction, and division are well defined & act as corresponding operations with real and rational numbers. This means that it is a set whose elements follow the +, -, x and / rule. The field is very important in Algebraic Number Theory and mathematics in general.
A finite field is a set with a defined number of elements. This means that it is a set whose elements are not moving towards infinity. A = [3, 6, 9, 12] is a finite field as it has 4 elements which is all it comprises of.
Mathematically, the best fields, are the rational numbers and the entire field of real numbers. The complex number fields are also considered among the top fields in Algebraic Number theory. Other secondary fields that do not get as much representation in mathematics include fields of rational functions, algebraic number fields and, algebraic function fields & P-adic fields.
What is an Extension Field?
Extension fields also exist in Field Concept. Mathematically a Field Extension is possible if there are at least two pairs of fields B and C such that B is a subfield of C. in this case, C is the field extension because it houses B. the field extension logic can be explained in the following examples.
The binary operations of addition & multiplication can make existing rational numbers in a set become subfields of the numbers that will be formed once the operation is carried out. In the same vein, the field of complex numbers is a field extension of all fields of real numbers. This is because the complex number is made up of both real numbers and imaginary numbers. Additionally, Real numbers are extension fields of rational numbers.
Algebraic Number Field?
The Algebraic Number field, which is also known as the number field, is a type of extension field denoted as K of distinct rational numbers denoted as Q such that K/Q comes with a finite degree. In simpler terms, K is an extension field that comprises Q and still has a finite dimension if it is considered a Vector Space over Q.
The Algebraic Number Theory plays a special role in the Algebraic Number Theory. An example of the number field is outlined below:
The Gaussian Rational is an example of the Algebraic number field. The Gaussian Rational is one of the earliest of a field extension of the rational number. The Gaussian rational is denoted as Qi which is normally read as Q adjoined i. It is given as a +bi and is what is known as the complex number today. The a and b are normal rational real numbers which are the Q while i is the imaginary unit. The i2 represents – 1.
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) . (c + di) = (ac – bd) + (ad + bc)i
What is a Ring?
Rings are considered algebraic structures that serve as a generalization of fields. In rings, the multiplication operation must not be commutative, and the multiplication inverses do not need to exist. A ring is a set that satisfies the addition and multiplication operations for integers. The elements in a ring may not just be an integer and may include complex numbers. Rings may also include objects that are not numbers, but for the focus of this piece, the scope will be limited to Ring relationships with integers and complex numbers.
In Ring, the multiplication operation is associative and distributive over the addition operation and there is also the multiplication identity element. This means that the rings satisfy the following axioms which are given as
The Ring (R) is an abelian group based on addition operation which means that
- (a + b) + c = a + (b + c) where a, b, & c are elements of real numbers (This is Associativity)
We can try this with an example where a = 3, b = 5, and c = 9
(3 + 5) + 9 = 3 + (5 + 9)
(8) + 9 = 3 + (14)
17 = 17
This prove the Associativity of the ring.
- a + b = b + a where a, & b, are elements of real numbers (This is commutativity)
Using the same values of a = 3, b = 5, and c = 9
3 + 5 = 5 + 3
8 = 8
This prove the commutativity of the ring.
- An element 0 exist in R such that a + 0 = a where a is an element of real number (This is the additive identity)
Let a = 3
3 + 0 = 3
This prove the additive identity of the ring.
- For any a in the real number, – a also exists such that a + (- a) = 0 (- a is considered a additive inverse)
Let a = 3
Then 3 + (- 3) = 0
This prove the additive inverse of the ring.
The Ring (R) is a Monoid based on multiplication operation, which means that
- (a . b) . c = a . (b . c) where a, b, & c are elements of real numbers (This is Associativity)
Let a = 3, b = 5, and c = 9
(3 . 5) . 9 = 3 . (5 . 9)
(15) . 9 = 3 . (45)
135 = 135
This prove the Associativity of the ring.
- An element defined as 1 exist in the Ring R that makes a . 1 = 1 . a where a stands for whatever number in the R (This is the multiplicative identity)
Let a = 4
4 . 1 = 1 . 4
4 = 4
This prove the Multiplicative identify of the ring
The Multiplication operation is totally distributive over the addition operation for the Ring which means that
- a . (b + c) = (a . b) + (a . c) where a, b, & c are elements of real numbers (left distributivity)
let a = 3, b = 5, and c = 9
3 . (5 + 9) = (3 . 5) + (3 . 9)
3 . 14 = 15 + 27
42 = 42
This proves the Multiplicative left distributivity of the Ring with respect to the addition operation
- (b + c) . a = (b . a) + (c . a) where a, b, & c are elements of real numbers (right distriutivity)
(5 + 9) . 3 = (5 . 3) + (9 . 3)
(14) . 3 = (15) + (27)
42 = 42
This proves the Multiplicative right distributivity of the Ring with respect to the addition operation
This piece has considered the Algebraic number theory in its basic form. The Ring is one of the most important concepts and every other definition that followed before it was to help students understand what it is. Understanding the Algebraic Number Theory further than has been considered requires going into deeper scope which is available in textbooks.