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Jan
20
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
I'm sorry again. I failed to post the previous comment. If $p(t) \in k[t]$ is the irreducible polynomial of $\alpha_1$ and $\tau_1:k(\alpha_1)\rightarrow L$ is the extension of $\sigma$, and $\tau_2:k(\alpha_1,\alpha_2)\rightarrow L$ is the extension of $\tau_1$, then the destination of $\alpha_2$ must be a root of $p^\sigma$. For $0=\tau_2(p(\alpha_2))=p^\sigma(\tau_2(\alpha_2))$.
Jan
20
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
I'm sorry. I assumed that $\beta_1$ is the destination of $\alpha_1$. If $\tau_2:k(\alpha_1,\alpha_2)\rightarrow L$ is the extension of $\tau_1:k(\alpha_1)\rightarrow L$.
Jan
20
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
When you extend $k(\alpha_1)\rightarrow L$ to $k(\alpha_1,\alpha_2)\rightarrow L$, you have to choose the destination of $\alpha_2$. If $\alpha_1 \neq \alpha_2$, that must be different from $\beta_1$.
Jan
20
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
It seems to me that to construct the embedding $\mu$ you need the same number of distinct roots of $p^\sigma(t)$ in $L$ as $p(t)$ in $k^a$. But that is what we want to prove in the first place.
Jan
19
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
Sorry for my repeated questions. When you extend the embedding $k(\alpha_1)\rightarrow L$ to $k(\alpha_1,\alpha_2)\rightarrow L$, you have to find another root of $p^\sigma$ in $L$. Is this always possible ?
Jan
18
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
Your argument is clear. I understand that the second part of the proposition is proved in that way. Do you think that it is impossible to prove it without using the extension of $\sigma$ to an embedding of $k^a$ in $L$ ?
Jan
18
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
Could you elaborate on the 4th part ?
Jan
18
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
This is exactly what is shown in Theorem 2.8. There is the following sentence imediately after the Proposition 2.7.: This is an important fact, which we shall analyse more closely later. Does this mean that the second part of the proposition will become clear later ? Or, is there other way to prove it ?
Jan
17
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
In Theorem 2.8. the existence of an extension of $\sigma$ to an embedding of arbitrary algebraic extension of $k$ is proved by using Proposition 2.7.
Jan
17
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
The edition is revised third.
Jan
17
asked The number of possible extensions of an embedding of a field into a algebraically closed field.
Jan
13
comment Triangle inequality of a metric on a quotient space of a topological vector space
inf A + inf B = inf (A+B). See for example Exercise 1.3.9. of understanding analysis by Stephen Abbott.
Dec
7
comment Question about extensions of homomorphisms
Thank you for answering my question. Am I right in thinking that an extension of $\psi$ is defined as a unique homomorphism of $A/\mathfrak{m}$-algebras $\tilde{\psi}:B/\mathfrak{P} \rightarrow L$ such that $\tilde{\psi}(x^{-1}+\mathfrak{P})=\lambda$ for any $\lambda\in L$.
Dec
6
asked Question about extensions of homomorphisms
Nov
30
answered Triangle inequality of a metric on a quotient space of a topological vector space
Nov
25
revised Triangle inequality of a metric on a quotient space of a topological vector space
added 7 characters in body
Nov
25
awarded  Editor
Nov
25
comment Triangle inequality of a metric on a quotient space of a topological vector space
@joriki, I'm sorry. X is a topological vector space. I corrected the question.
Nov
25
revised Triangle inequality of a metric on a quotient space of a topological vector space
added 7 characters in body
Nov
25
asked Triangle inequality of a metric on a quotient space of a topological vector space