# Splitting $f(x)$ in $K_n$ Extension Field. What is the maximum degree of $K_n$

Assume $$f(x) \in F[x]$$ has degree $$n$$ and there exists a field extension that splits $$f(x)$$, that is

$$f(x)=c_0(x-c_1)\dots (x-c_n)$$

Prove that the degree of the field extension of $$F$$ is at most $$n!$$

Assume $$f(x) \in F[x]$$ has degree n.

Step 1

Worst case scenario $$f(x)$$ is irreducible in $$F[x]$$. We generate $$F[x]/(f(x))=K_1$$ where $$c_1=[x]$$ is a root. So in $$K_1[x]$$ $$f(x)=(x-c_1)g_1(x)$$ Step 2 Worst case scenario again $$g_1(x)$$ is Irreducible. We generate $$K_2=K_1/(g_1(x))$$ so their is a root $$c_2$$ in K_2 where $$f(x)=(x-c_1)*(x-c_2)g_2(x)$$ $$\vdots$$

After a maximum of $$n$$ steps

$$\exists K_n[x]$$ where $$f(x)=c_0(x-c_1)\dots (x-c_n)$$

Note that the important thing is that $$F[x] \subset K_1=F[x]/(f(x)) \subset K_2= (F[x]/(f(x)))/(g_1(x)) \subset \dots \subset K_n$$ We want to find the maximum degree of $$K_n$$ over $$F$$, that is, $$[K_n:F]$$. Using the tower law, $$[K_n:F]=[K_n:K_{n-1}]\dots [K_i:K_{i-1}]\dots [K_1:F]=1*2*3* \dots* (n-1)*n$$ Since $$[K_i:K_{i-1}]=deg (g_i(x))$$ that is the degree of irreducible that was used to generate the new extension field. Each extension field is $$1$$ degree lower so that is why it is a factorial $$n!$$ at the most

Questions I assumed it is irreducible I believe it should say doesn't have a root but the field generated would not be a field. Did not stated what happens when it was reducible. Also is a chunk of it a proof or close to a prove that $$f(x)$$ can be split in some extension field. I know induction is strongly preferred.

## 1 Answer

If $f$ is reducible, say $f(x)=h(x)g(x)$ where $h(x)$ is irreducible, then you can use a similar argument when adding a root of $h$ to the field. Note that it is just fine even if $f(x)$ has a root (in this case, $h(x)=x-a$ and adding $a$ simply wont extend the field).

If you could show how to add a root to a field, without assuming a containing splitting field, then your argument could extend to show the existence of a splitting field (it is not unique by the way!). Hint: consider a quotient of $K[x]$.