Algebra is maybe my favorite area of math to study. Right now I’m trying to learn more linear algebra and category theory. Linear algebra is one of the most powerful and applicable areas of math today, as statistics, machine learning, and AI heavily depend on it. Category theory is notoriously difficult to describe, but it is fun because of how many layers of abstraction you have to deal with. It is truly mind bending. Solving problems in algebra amounts to playing with and manipulating the definitions and theorems until you get what you want. Algebra goes far beyond what you learn in high school, and you can do algebra with much more than just numbers. Here I want to write about two different ways to think about algebra with polynomials.

Polynomials

Firstly, a polynomial is a function that can be written as

Note that some of the $a$’s can be 0, cancelling out that power of $x$. The maximum power of $x$ is called the degree of the polynomial. I will write $\mathcal{P}(x)$ to represent all polynomials, and $\mathcal{P}_{m}(x)$ to represent all polynomials with degree at most $m$. Now we can see how we can do algebra with these polynomials.

Polynomials as a vector space

A vector space is a set (here, $\mathcal{P}(x)$), where you can add and scale all the elements (along with a few more constraints, scroll to the end). An element of a vector space is called a vector. So, given any polynomials, like $f(x)=1+x^{2}$ and $g(x)=3x^{2}-5x^{4}$, we can compute

and scaling like

So you can see how we can perform addition on polynomials, just like how we can add two numbers. Note that here we are not worried about actually plugging in any value for $x$ and evaluating, our “numbers” are the polynomials themselves.

We can also form a vector space with $\mathcal{P}_{m}(x)$. It is clear that you cannot get a polynomial with degree greater than $m$ from addition and scalar multiplication of polynomials with maximum degree $m$. For example, $\mathcal{P}_{3}(x)$ is all the polynomials with at most degree 3. You cannot get a polynomial of degree 4 from addition of two polynomials of degree 3.

You can get any polynomial in $\mathcal{P}_{3}(x)$ from addition and scalar multiplication of the polynomials

This is called the basis of the vector space. Since there are 4 polynomials in the basis, the dimension of $\mathcal{P}_{3}(x)$ is 4. In general, the dimension of $\mathcal{P}_{m}(x)=m+1$. $\mathcal{P}(x)$ is infinite-dimensional.

Polynomials as a ring

A ring is a set (again, here the set of polynomials), where you can add and multiply elements together. Additionally, here we are only considering polynomials with integer coefficients. Addition is defined as before, and multiplication is similar

You can see that here in the ring of polynomials, once we have a polynomial with degree at least 1, we can multiply it by itself to get a polynomial with an arbitrarily high degree. So $\mathcal{P}_{m}(x)$ cannot be a ring, since we can “get outside” of the ring by combining elements inside of it (this wasn’t possible in a vector space since multiplication between two vectors isn’t defined). Since $f(x)=1$ multiplied by any polynomial equals $1$, $1$ is a unity of the ring of polynomials with integer coefficients. Also, the ring of polynomials is an integral domain since if $a,b$ are polynomials and $a\cdot b=0$, then $a=0$ or $b=0$.

Conclusion

Algebra is much more expansive than what is taught in high school. Here, you can see how you can define addition and multiplication to work with polynomials, and then you can do algebra with polynomials in much the same way you do algebra they way you are used to with numbers. There are many more algebraic structures than the two I listed here, such as groups, fields, and monoids, and you can do algebra on elements that are numbers, polynomials, functions, rotations, letters, and more. Studying algebra abstractly lets you see what all of these have in common, without getting lost in the details of a particular realization of the structure.

Formal definitions

Vector Space

A vector space is a set $V$ (here, let $u,v,w\in V$) along with addition and scalar multiplication such that

1. $u + v = v + u$ (addition is commutative)
2. $(u + v) + w=u+(v+w)$ (addition is associative)
3. there exists an element $0\in V$ such that $v+0=v$ for all $v\in V$ (there is an additive identity)
4. for every $v\in V$, there exists $w\in V$ such that $v+w=0$ (additive inverses exist)
5. $1v=v$ for all $v\in V$ (there is a scalar identity)
6. $a(v+w)=av+aw$ and $(a+b)v=av+bv$ for all $a,b\in \mathbf{F}$ (scalar multiplication distributes over addition)

$\mathbf{F}$ is any field where the scalars come from, and you say $V$ is a vector space over $\mathbf{F}$.

Ring

A ring $R$ is a set with addition and multiplication such that

1. $a+b=b+a$ (addition is commutative)
2. $(a+b)+c=a+(b+c)$ (addition is associative)
3. there exists an element $0\in R$ such that $a+0=a$ for all $a\in R$ (there is an additive identity)
4. for all $a$, there exists an element $-a\in R$ such that $a+(-a)=0$ (additive inverses exist)
5. $a(bc)=(ab)c$ (multiplication is associative)
6. $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$ (multiplication distributes over addition)

Note that multiplication in a ring does not necessarily have to be commutative, have inverses, or have an identity.