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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 $F$.

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  • $\begingroup$ 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. $\endgroup$ Nov 25 at 10:16
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From the definition of one-to-one mapping of a function, for part (a): Let a₀=b₀,a₁=b₁,...,aₙ=bₙ ∈ F, then ι(a₀,a₁,...,aₙ) = (a₀ + a₁x + ... + aₙxⁿ) + <p(x)> and ι(b₀,b₁,...,bₙ) = (b₀ + b₁x + ... + bₙxⁿ) + <p(x)>. Thus, ι(a₀,a₁,...,aₙ) = ι(b₀,b₁,...,bₙ). Hence, ι is one-to-one. (Note that F[x]/<p(x)> is the notation for a quotient ring, also known as factor ring, which is a set of cosets and whose elements are f(x) + <p(x)>, f(x) ∈ F[x].)

hopefully next post for part(b)

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