23
votes
6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
4
votes
2answers
124 views

Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
1
vote
1answer
69 views

Mathematics and Origami

I am reading through this paper about the math behind origami: http://www.math.washington.edu/~morrow/336_09/papers/Sheri.pdf However, I am getting confused with definitions 3.3 and 3.4. I am not sure ...
3
votes
2answers
114 views

Math and Origami

I am working on a project for class about the mathematics behind origami and write now I am looking into what is and is not constructible. I've gotten to the definition of origami constructible points ...
1
vote
1answer
86 views

Is $\Bbb Q[\sqrt2]$ cyclotomic?

This overview of Galois Theory claims that a field extension of $F$ is cyclotomic if it's obtained by adjoining an $n$th root of any element of $F$. Wikipedia claims you have to adjoin a root of unity ...
3
votes
2answers
182 views

Understanding of extension fields with Kronecker's thorem

In the book Contemporary Abstract Algebra by Gallian it defines an extension field as follows: A field $E$ is an extension field of a field $F$ if $F\subseteq E$ and the operations of $F$ are ...
2
votes
1answer
150 views

What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
3
votes
1answer
513 views

Definition of Separability Degree

For an assignment, I am trying to determine the separability degree of some algebraic field extension $L/K$. The definition of the separability degree of polynomial is not difficult to find at all, ...
2
votes
2answers
1k views

What is Galois Field

When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in ...
0
votes
1answer
221 views

Principle of substitution — fields

Could someone possibly explain this definition (applied to fields) to me? The principle of substitution: In a field F, we can, in any formula involving an element $\alpha\in F$, replace $\alpha$ ...