Reminder: This post contains 1677 words
· 5 min read
· by Xianbin
It is very important to choose the right book to learn something new.
When I read the book [1], I feel someone is pushing his/her wisdom to my head. I strongly recommend someone who is new to abstract algebra reads this book[1].
Definition of Field
I assume readers of this blog know basics about set theory.
We say that a non-emptyset \(G\) is a field if in \(G\), there is an operation * such that:
- \(a, b \in G\) implies taht \(a*b \in G\) (closure).
- \(a, b, c \in G\) then, \(a*(b*c) = (a*b)*c)\) (associativity)
- There exists an element \(e\in G\) such that \(a*e = e*a\) for all \(a\in G\)
- For every element \(a \in G\), there exists an element \(b\) such that \(a*b = b*a = e\).
See, it is simple and elegant.
If the number of element in \(G\) is finite, we say that it is a finite group.
Now, we always see that if \(a*b = b*a\) for all \(a,b \in G\), it is called abelian group.
Examples
1). Let \(*\) be the ordinary addition and \(Z\) the set of integers. Then, \(Z\) is a group.
2). Let \(E_n = \theta_n = \text{cos}(2\pi /n) + i \text{ sin}(2\pi/n)\) and \(*\) be the ordinary product. It is a finite group.
Quick question:
Can you give me an example in which it does not satisfies abelian?
Apparently, we should construct some operations that do not look very symmetrical? Like matrix multiplication. Now, let us see an example.
Let \(R\) be the set of all real numbers and we define \(G\) to be the set of all mapping \(X_{a,b}: R\to R\): \(X_{a,b}(r) = ar+b\) where \(a,b\) are real numbers and \(a \neq 0\).
Consider two mappings \(X_{a,b}\) and \(X_{c,b}\). By the definition, we have
\[X_{a,b} * X_{c,d} = X_{ac,ad+b}.\]
By checking the above four properties, we can see that it is a group.
Now, let us see \(X_{c,d} * X_{a,b}\). (just exchange the letters).
\[X_{c,d} * X_{a,b} = X_{ca,cb+d}.\]
Apparently, it is not abelian.
If we consider \(X_{1,b}\in G\), then it is abelian. The reason is simple, replace \(a,c\) by 1, we get the same result.
Reference
[1]. Herstein, I.N., 1996. Abstract algebra. John Wiley & Sons.