Why is abstract algebra named algebra?

Question

Why is abstract algebra named like the algebra we see in high school ? They have nothing in common. Also, why are group, rings and fields named algebraic structure ?

Answer 1

Originally algebra was merely the symbolic manipulation of quantities. So representing mathematical things with letters and other symbols like $+$ and $=$ makes it algebra. We're just moving symbols around using certain rules that normal quantities obey. These quantities were typically elements of a field and had all the nice properties that fields come with.

However abstract algebra discards the necessity that the symbols have a meaning, which is what makes it abstract. We can just make symbols and give them rules to manipulate expressions. This lets us generalize and build a mathematical theory without appealing to things like numbers.

Starting with the field axioms a natural question is does it have any substructure? For example the integers are a substructure of the rational numbers but are not a field, because they lack multiplicative inverses, but we can still do a lot of algebra with them. The natural numbers aren't even a group but I can still do symbolic manipulation regarding natural numbers even though they're just a monoid. This leads us to relaxing as many axioms as we can. Groups, rings and fields have all proven useful in various contexts.

Groups are the only one you haven't really encountered since the integers are a ring but groups are not always assumed to be commutative. I think the reasons groups are studied so much is that they have many applications. For example in crystallography we can reconstruct the shape of a molecule by examining the symmetries present in the diffraction patterns. This is how DNA was determined to be a helix as it had certain periodic and rotational symmetries. Even macroscopic structures like snowflakes reveal important details about the molecular structure by examining how groups act on their shape.

I hope that gives you some understanding as to why abstract algebra is actually a kind of algebra.

Answer 2

By the way: the word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting' from the title of the early 9th century book cIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi.

Answer 3

I'll give you an example. A group is an algebraic structure. It is a set coupled with an operation, where the operation takes in two inputs and gives one output. A group has the following properties:

1. The operation follows the associative property.
2. The operation has an identity that is in the set. If A is the operator and x is from the set, then there is an I from the set such that $A(x,I) = x$. Like for addition, the identity would be $3 + 0 = 3$.
3. The operation has closure over the set. So any two inputs from the set will give an output also in the set.
4. The operation has an inverse that also has closure over the set and results in the identity element. For example, $3 * \frac{1}{3} = 1$, which is the multipilcative ideneity.

For example, the real numbers with addition are a group. Let's look at an example of how this relates to Algebra 1: \begin{align*} (x + 3) &= 5 \\ (x + 3) + -3 &= 5 + -3 &\text{[inverse]} \\ x + (3 + -3) &= 5 + -3 &\text{[associative]} \\ x + 0 &= 5 + -3 \\ x &= 5 + -3 &\text{[identity]} \\ x &= 2 &\text{[closure]} \end{align*}

Answer 4

This statement is not correct, abstract algebra provides definitions and generalizes everything you learned in high school. Perhaps your professor failed to make these connections sufficiently clear. And an algebraic structure is a set + operation(s) + rule(s), it seems like a fairly straightforward name.