Number Theory and Abstract Algebra: Foundations of Pure Mathematics, Study Guides, Projects, Research of Abstract Algebra

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Number Theory & Abstract Algebra

Mathematics Informatics

Informatics Engineering

Gunadarma University

Overview Number Theory and Abstract Algebra are foundational areas of pure mathematics. Number Theory deals with the properties and relationships of numbers, especially integers. Abstract Algebra studies algebraic structures such as groups, rings, and fields. Key Concepts in Number Theory

1. Divisibility and Prime Numbers - Divisibility : An integer a divides b if there is an integer k such that b = ak. - Prime Numbers : A prime number is greater than 1 and has no positive divisors other than 1 and itself. o Example: 2, 3, 5, 7, 11, etc. - Greatest Common Divisor (GCD) : The largest positive integer that divides two integers without leaving a remainder. o Euclidean Algorithm: Used to compute the GCD of two numbers. - Least Common Multiple (LCM) : The smallest positive integer that is divisible by both of two given numbers. 2. Congruences - Modular Arithmetic : Numbers are said to be congruent modulo nnn if they have the same remainder when divided by n. a ≡ b (mod n ) if n divides ( a – b ) - Chinese Remainder Theorem : Provides a unique solution to a system of simultaneous congruences with pairwise coprime moduli.

2. Rings - Ring : A set R equipped with two binary operations (usually addition and multiplication) satisfying: o Additive Identity : There exists an element 0 ∈ R such that a + 0 = a for all a ∈ R. o Additive Inverse : For each a ∈ R , there exists an element − a ∈ R such that a + (− a ) = 0. o Multiplicative Identity (optional): There exists an element 1 ∈ R such that a ⋅ 1 = a for all a ∈ R. o Distributivity : a ⋅ ( b + c ) = a ⋅ b + a ⋅ c for all a , b , c ∈ R. 3. Fields - Field : A ring F where every non-zero element has a multiplicative inverse. o Example: The set of rational numbers ℚ, real numbers ℝ, and complex numbers ℂ. Example Problem in Abstract Algebra Problem : Determine if the set of integers ℤ with addition and multiplication forms a ring. Solution : 1. Check properties of a ring: o Additive Identity : 0 is the additive identity in ℤ. o Additive Inverse : For each a ∈ ℤ, − a is in ℤ and a + (− a ) = 0. o Multiplicative Identity : 1 is the multiplicative identity in ℤ. o Distributivity : For all a , b , c ∈ ℤ, a ( b + c ) = ab + ac. 2. Conclusion: ℤ with standard addition and multiplication forms a ring.

Conclusion Number Theory and Abstract Algebra are crucial for understanding the properties of numbers and algebraic structures. By studying key concepts and practicing problems in divisibility, congruences, Diophantine equations, group theory, ring theory, and field theory, you will build a strong foundation in these areas. Practice Problems Number Theory

1. Find the LCM of 15 and 20.
2. Solve the congruence 7 x ≡ 3 (mod 10)
3. Find all integer solutions to the Diophantine equation 3x + 4y = 7. Abstract Algebra
4. Show that the set of 2x2 matrices with real entries forms a ring under matrix addition and multiplication.
5. Prove that the set of non-zero rational numbers forms a group under multiplication.
6. Determine if the set of polynomials with real coefficients forms a ring.