Show that $K[x]/(x)$ is isomorphic to $K$, where the map is defined by sending a polynomial to a constant coefficient

Question

I have a problem here.

show that $K[x]/(x)$ is isomorphic to $K$, where the map is defined by sending a polynomial to a constant coefficient.

My attemot on this problem is to defined a function $f: K[x]/(x) \rightarrow K$. And then show that $f$ is isomorphic. However, I don't how the elements of the domain looks like. Can somebody help me? I will turly appreciate it.

Answer 1

Welcome to MSE! You're most of the way there.

Elements of $K[x]$ are polynomials with coefficients in $K$. Knowing this, you already have a function

$$a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n \longmapsto a_0 : K[x] \to K$$

which takes a polynomial as input and outputs the constant term of that polynomial. For instance, we have things like $f(x^2 + 1) = 1$ and $f(9x^4 - 2x^2 + 11x - 3) = -3$ and so on.

Do you see why this map is surjective? Do you see what its kernel is? With these facts together, can you use the first isomorphism theorem to get the isomorphism you want?

---

I hope this helps ^_^