This is a note about abstract algebra, one of the largest branches of mathematics. I will show some concepts in abstract algebra and the power of abstraction.
What is abstraction?
Before getting our hand's dirty with some algebra, first let us consider the concept of abstraction. Abstraction is, loosely speaking, the process of taking a problem or a concept and removing unnecessary context. This is an important concept in mathematics. It is what allowed us to view the number for example, as an abstract concept rather than lots of sheep. Another context we could think about is shapes. Let's say we know a fact about a triangle and a square. We later do some observation to reveal that this also applies to a pentagon and hexagon and so on. However how do we prove that this is the case? Well we cannot prove it for every polygon we can come up with because there are infinitely many! But what if we proved the hypothesis for an -gon. Independent of the number of vertices. Then we have automatically proven it for all polygons! This is the power of abstraction.
Sets are arguably the most important concepts in Mathematics. All of Mathematics can be derived from sets! However i will not show this here. I am interested in some special sets. These sets are used on a daily basis and by explaining the structures they possess, it helps give a better understanding of the. Firstly let us look at some commonly used sets:
You are probably familiar with these sets, if you are not then this post will not make much sense to you. One thing to note is that all of these sets share many algebraic properties. These properties define an abstract object called a ring.
Definition 1 A ring is a set of elements on which two binary operations, addition and multiplication , are defined that satisfy the following properties for all :
Note that multiplication in a ring is not always commutative. A ring where multiplication is commutative is called a commutative ring. An example of a non-commutative ring would be , the collection of matrices with integer entries. Remember, matrix multiplication is not commutative.
Already we can prove some things about rings,. Suppose is a ring and then:
These properties of rings are not in the definition but derive directly from the ring definition. It is fun to prove these however remember that is not necessarily commutative so the order of multiplication cannot be switched.
Now let us look at more concepts about rings.
A subset of a ring is said to be a subring of if is itself a ring under the operations induced from .
This is an important concept because it tells us that ring structure can be inherited for example and both are rings therefore is a subring of .
The Subring theorem A non-empty subset of a ring is a subring under the same operations if and only if it is closed under multiplication and subtraction.
This is an important theorem because it allows us to check if a subset is indeed a subring without checking every property of a ring. This can be proven however it will not be proven here.
Some rings have a unity, or multiplicative identity; that is, an element where for all . If a unity exists, it is unique.
For example the number is the unity in , , and . The matrix is the unity in ). But , the set of even integers , has no unity.
An element is a zero divisor if there is an element with . For example in
A commutative ring with unity is an integral domain if it has no zero divisors. , , and are all integral domains. The integers modulo , , is only an integral domain if and only if is prime.
Integral domains have the nice property of multiplicative cancellation. We can state this as a theorem:
If is an integral domain and with , then implies that .
If is a ring with unity , then an element is a unit if there exists a such that . In this case, is said to be the multiplicative inverse of . If all the non-zero elements of a commutative ring with unity are units, then we say the ring is a field. The rings , and are all fields but is not. All fields are integral domains. is a field, for a prime . Indeed, all finite integral domains are fields. The fact that is a field is an important setting for proving Fermat's little theorem which states that:
If is prime and , then .
Finally, consider , the set of polynomials over an arbitrary field with coefficients in . is an integral domain. The Division Theorem, the Root Theorem, Euclid's Algorithm, the GCD identity and Unique Factorization all hold for . A particularly important example is , the field of complex numbers. satisfies the Fundamental Theorem of Algebra: Every non-constant polynomial in has a root.
There are a lot of theorems mentioned earlier. You may recognise some of them but as you can see the power of abstract algebra has generalised them for the field of polynomials rather than for integers.
I hope this post has enlightened you about the power of abstract algebra and given you a taste of higher Mathematics. The last paragraph about polynomials over fields is an interesting thing that has many fun proofs to try. But an interesting note to take is that all of these things that we do with integers, arithmetic, GCD and factorisation and so on can be generalised to polynomials and other sets.