Mathematics

Understanding the Mathematical Subject Abstract Algebra



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Abstract Algebra has to do with systems of algebraic equations. Beginning Abstract Algebra is based on the book Euclid's Elements. The Euclidean Algorithm and Euclid's proof that the prime numbers are infinite are important. The Euclidean Algorithm is a method that finds the greatest common divisor between two integers using the division algorithm. The greatest common divisor is the greatest integer that divides two integers. For example, the greatest common divisor between 24 and 36 is 12 because since 24=2x2x2x3 and 36=2x2x3x3, the greatest integer between the two is 2x2x3=12. If two integers have a greatest common divisor of 1, then they are called relatively prime. Examples are 7, 12; 15,38; etc. An integer is prime if its only divisors are 1 and itself.
An integer that is not prime is composite. The prime factorization can be used to determine if two integers are relatively prime. Using the examples, 7=7x1, 12=2x2x3; 15=3x5, 38=2x19. Prime numbers are always relatively prime.

The Fundamental Theorem of Arithmetic is the fact that there is only one prime factorization of every integer. This is because prime numbers cannot be divided by any integer except 1 and itself. Examples of prime factorizations are 100=2x2x5x5, 35=7x5, 125=5x5x5, etc. Computers are being used to find
the prime factorization of very large integers with thousands of digits. The Fundamental Theorem of Algebra claims that the number of roots of a polynomial is the degree of that polynomial. For example, the number of roots of a linear equation is 1 and a quadratic equation is 2. The Fundamental Theorem of Calculus is the definition and properties of the integral.

Congruences are also important. The terminology is a is congruent to b mod n if n divides a-b. Divides means the quotient (a-b)/n does not have a remainder. For example, 7 is congruent to 5 mod 2 because 7-5=2 and 2/2=1. Several rules concerning congruences can be used. One is that if
a is congruent to b mod n, then b is congruent to a mod n. In our example, 5-7=-2, -2/2=-1.
Another is if a is congruent to b mod n and b is congruent to c mod n, then a is congruent to c mod n.
An example is 8 is congruent to 4 mod 2 and 4 is congruent to 2 mod 2, so 8 is congruent to 2 mod 2.
This is true because (8-4)/2=4/2=2, (4-2)/2=2/2=1, and (8-2)/2=6/2=3. Also if a is congruent to b mod n,
c is congruent to d mod n, then a+c is congruent to b+d mod n and axc is congruent to bxd mod n.

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