Show that the mapping is surjective

In summary, the conversation discusses the application of a theorem to show the existence of a monomorphism that maps one element in an algebraic closure to another, and how this monomorphism is onto and an automorphism. The conversation also raises questions about the monomorphism's behavior with respect to the field and whether it preserves algebraic closure.
  • #1
mathmari
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Hey! :eek:

I am looking at the following exercise:
Let $C$ be an algebraic closure of $F$, let $f\in F[x]$ be irreducible and let $a,b\in C$ be roots of $f$.

Applying the theorem:
"If $E$ is an algebraic extension of $F$, $C$ is an algebraic closure of $F$, and $i$ is an embedding (that is, a monomorphism) of $F$ into $C$, then $i$ can be extended to an embedding of $E$ into $C$. "

show that there is a $F$-monomorphism $\tau :C\hookrightarrow C$, that maps $a$ to $b$ and how then that $\tau$ is onto, so it is an automorphism of $C$. I have done the following:

We have that $C$ is an algebraic closure of $F$ and $a$ and $b$ are elements of $C$, so $C$ is also an algebraic closure of $F[a]$ and $F$.
Then we have the algebraic extensions $C\leq F[a]$ and $C\leq F$.
Since $F[a]$ and $F$ are subfields of $C$, there are the field monomorphisms $\tau_{a}: F[a]\rightarrow C$ and $\tau_{b}: F\rightarrow C$.
We have that $f\in F[X]$ is irreducible, then there is a field that contains $F$, in which $f(X)$ has a root.
This field is $K=F[X]/\langle f(X)\rangle$.
We have that $K=F[a]$ and $K=F$.
So, we have that $F[X]/\langle f(X)\rangle=F[a]$ and $F[X]/\langle f(X)\rangle=F$.
Therefore, we have the field monomorpshims $\tau_{a}: F[X]/\langle f(X)\rangle\rightarrow C$ with $\tau_{a}(g(X))=a$ and $\tau_{b}: F[X]/\langle f(X)\rangle\rightarrow C$ with $\tau_{b}(g(X))=b$.
Applying the above theorem with $E:=C$ and $F:=F[X]/\langle f(X)\rangle$, we have the monomorphism $\tau : C\rightarrow C$ with $\tau \circ \tau_{a}=\tau_{b}$.
So, $\tau ( \tau_{a}(g(X)))=\tau_{b}(g(X)) \Rightarrow \tau ( a)=b$.

Is this correct? (Wondering) From $\tau:C\rightarrow C$ we have that $\tau (C)\subseteq C$.
We have that $F\leq C\Rightarrow \tau (F)\subseteq \tau (C)$.
Since $\tau$ is a $F$-monomoprhism, do we have that $\tau (F)=F$ ? (Wondering)
 
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  • #2
We have that $\tau :C\rightarrow C$ maps $a$ to $b$. Since $F\subseteq C$ and $a,b\notin F$, we have that $\tau (f)=f, \forall f\in F$, or not? (Wondering)
So, $\tau (F)=F$.
Is this correct? (Wondering)
 
  • #3
Do we have that $\tau (C)$ is algebraically closed? If so, why? (Wondering)
 

FAQ: Show that the mapping is surjective

What does it mean for a mapping to be surjective?

A surjective mapping is one in which every element in the codomain has at least one corresponding element in the domain. In other words, every element in the output has at least one input value that maps to it.

How is surjectivity different from injectivity?

Surjectivity and injectivity are two different properties of a mapping. While surjectivity ensures that every output has at least one corresponding input, injectivity ensures that every output has at most one corresponding input.

How is surjectivity demonstrated or proved?

In order to prove that a mapping is surjective, you must show that for every element in the codomain, there exists at least one element in the domain that maps to it. This can be demonstrated through examples or using mathematical proofs.

What are some common examples of surjective mappings?

Some common examples of surjective mappings include the mapping of real numbers to their absolute values, the mapping of integers to their square roots, and the mapping of natural numbers to their cubes.

What is the importance of surjectivity in mathematics and science?

Surjectivity is an important property in mathematics and science because it ensures that all elements in the output have a corresponding input, making the mapping a more complete and useful representation of the relationship between two sets of data.

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