Difference between $k[X]$ and $k(X)$ where $k$ is a field

Question

Can someone please explain me elaborately what is the difference between $k[X]$ and $k(X)$ where $k$ is a field?

Answer 1

$k[X]$ is the ring of polynomials in $X$, so its elements are of the form $$a_0 + a_1X + a_2X^2 + ... + a_nX^n$$ with $a_i \in k$ and $n$ a non-negative integer. The operations are addition and multiplication of polynomials, and addition is inverted by subtraction.

On the other hand, $k(X)$ is the field of rational expressions in $X$, so its elements are quotients of elements in $k[X]$. This can be thought of as the field of fractions of $k[X]$, so that multiplication is now invertible.

Answer 2

$k[x]$ are polynomials. This is a ring, and it is easy to check that since $k$ is a field the only invertible elements in $k[x]$ are the constants. On the other hand, $k(x)$ is a field; it consists of all rational functions.

Answer 3

I always thought of it as k[X] is the smallest ring extension of k containing X and since X has no specific relation with the elements of k, its elements are in the form of a polynomial aX^n+bX^(n-1)... where a,b... are in k.

k(X) is the smallest field containing X also called field generated by X. its like k[X] except you can divide by elements of k[X]; it is the field of fraction.

A fun fact I was surprised to learn was that if X is a finite extension, so there is a finite polynomial in X that is always 0, then k[X] is already a field and k[X]=k(X). Pretty useful since most extensions I need to know about are finite.