# Mapping from $F$ to $E$ where $F$ is a field and $E=F[x]/\langle p(x)\rangle$

Here is the question as given that I am having trouble with:

Let $$F$$ be a field and $$p(x)$$ an irreducible polynomial in $$F[x]$$. In this investigation we showed that $$E=F[x]/\langle p(x)\rangle$$ is a field, and we implied that $$F$$ is a subfield of $$E$$. Now we will examine what we mean by that statement.

(a) There is a natural mapping $$\iota$$ from $$F$$ to $$E$$. Identify this mapping ($$\iota$$ is called the inclusion mapping). Show that $$\iota$$ preserves the structure of $$F$$. Is $$\iota$$ an isomorphism? Explain.

(b) Explain how $$E$$ contains an isomorphic copy of $$F$$. (It is in this sense that we say $$F$$ is a subfield of $$E$$. This subfield of $$E$$ this is isomorphic to $$F$$ is called an embedding of $$F$$ in $$E$$.)

Keep in mind that I am really new to all of this so I am trying my best to make sense of a ton of new information. I think I understand the idea of quotients with polynomial rings and why $$E=F[x]/\langle p(x)\rangle$$ is a field, but I am unclear what part (a) is looking for in terms of the mapping. We have also talked about an isomorphism being a well-defined bijective function that preserves addition and multiplication. Is this what I would be checking to show that the mapping is an isomorphism then? For part (b), I'm not sure why exactly $$E$$ would contain an isomorphic copy of $$F$$.

• For part (b), you may try reading A first course in abstract algebra by Fraleigh, 7th Ed, Theorem 22.4 The Evaluation Homomorphisms for Field Theory, page 201 - 203. Change the domain from F[x] to F[x]/<p(x)>, the theorem is probably still valid. Nov 25 at 10:16