The set of integers is another name for the set of whole numbers; in other words the set {...-2, -1, 0, 1, 2, ...}. The set of integers is commonly denoted by the symbol Z. Under the familiar operations of addition and multiplication on whole numbers it is easy to show that the integers Z along with addition satisfy the axioms for an Abelian group, that is a commutative group, and that Z along with addition and multiplication form a commutative ring with identity.
Basic Algebraic Properties of the Integers
It is necessary to note that Z along with multiplication does not form a group since the element 0 in Z does not have a multiplicative inverse and thus fails to satisfy all of the group axioms; however the set Z\{0} along with multiplication does form a group, indeed, an Abelian group.
The (commutative) ring of integers is often considered as the most fundamental and natural ring. The importance of the ring of integers is reflected in the elementary theorem from ring theory which is not proved here, but which is as follows
Theorem: For an arbitrary ring R there is a unique ring homomorphism f : R --> Z
The Fundamental Theorem of Arithmetic
Theorem: Every integer greater than 1 can be decomposed as a product of prime numbers and such a decomposition is unique up to re-ordering of the terms.
Considering the first few cases we have
2=2
3=3
4=2x2
5=5
6=3x2
7=7
8=2x2x2
9=3x3
Can it indeed be true that every number can decompose in this way? The theorem can be proved using mathematical induction, but a much more succinct proof could go as follows:
With a small amount of work, it can be proved that for any two integers m and n (with n not equal to 0) it is possible to find unique integers q and r with 0 smaller than or equal to q, and r strictly smaller than the absolute value of n such that m=qn+r.
This is an example of a Euclidean function on Z, and as a result, Z is shown to be a Euclidean domain (ED). Since Z is an ED it follows that Z is a principal ideal domain (PID), which in turn implies that Z is a unique factorization domain (UFD) and the proof is complete since the above theorem is a special case of the definition of a UFD.
So, indeed it is true that every integer greater than 1 has a unique prime number decompositon.
Further Theory of the Integers
The integers may seem very simple at first sight. In some ways they are: they are the whole numbers; the numbers that are used to count; the first type of number that we encounter as children; but the theory surrounding the integers goes much deeper.
Above are just a handful of theorems and properties concerning the integers and especially the prime numbers. A conjecture put forward by the mathematician Bernhard Riemann in the 1800s concerning the distribution of prime numbers remains unsolved to this day. The statement of Fermat's last theorem, which involves powers of integers, is so simple that it can be understood by practically anyone; yet its proof eluded even the greatest of mathematicians until in the 1990s when Andrew Wiles finally produced a proof extending to dozens of pages. Entire books have been written describing the integers and their mathematical beauty; their importance in mathematics should not be overlooked.
Suggested Further Reading
Concrete Abstract Algebra by Niels Lauritzen (Cambridge University Press, 2005)
John Fletcher - I am a recent graduate of the University of Warwick in the UK, where in 2009 I received a first class master's degree in mathematics. My ...
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