Timeline for Why does the field norm on the field extension $ \mathbb C/\mathbb R $ induce a vector space norm?
Current License: CC BY-SA 4.0
15 events
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| Feb 17, 2022 at 13:11 | comment | added | Uri Bader | @DenisNardin hmm... thanks for noticing. Maybe I had something in mind when I wrote it, possibly a mistake. I will need to make a correction somewhere either way. I will look into it later on. | |
| Feb 17, 2022 at 11:23 | comment | added | Denis Nardin | @UriBader Ah I was confused by your remark in the other answer claiming that this proof works for all complete valued fields | |
| Feb 17, 2022 at 8:45 | comment | added | Uri Bader | @DenisNardin note that I have a standing assumption that $K$ is a local field here, so this answers your question. However, you are right asking it. I'd be happy to have a variant of the argument which works for arbitrary complete fields. | |
| Feb 16, 2022 at 22:05 | comment | added | Denis Nardin | @UriBader why is the unit ball in $K$ compact? I thought that was true iff $K$ is locally compact | |
| Aug 26, 2021 at 7:25 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 25, 2021 at 17:35 | comment | added | Ege Erdil | The argument now seems fine. I'll accept this answer if no simpler answer is given in the next day or two, but I'm still hoping that there's a simpler way to get this result... | |
| Aug 25, 2021 at 13:37 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 25, 2021 at 12:14 | comment | added | Uri Bader | The text is now complete, in my mind. I apologize for presenting a non-complete text earlier on. | |
| Aug 25, 2021 at 12:13 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 25, 2021 at 9:18 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 24, 2021 at 21:40 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 24, 2021 at 21:34 | comment | added | Uri Bader | Thanks for the notes: I was too brief and too sloppy. My answer describes a uniform proof that the intended norm is indeed a norm, in fact an absolute value. This follows from the fact that it is a $C$-ultra absolute value (which I didn't elaborate on much) plus the fact that every extension of a norm which is a $C$-ultra absolute value is an actual absolute value, as we have $|2|\leq 2$. | |
| Aug 24, 2021 at 21:30 | history | edited | Uri Bader | CC BY-SA 4.0 |
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| Aug 24, 2021 at 21:28 | comment | added | Ege Erdil | The question is specifically about completions of $ \mathbb Q $ because it's not true in general that this expression induces a norm on the extension. For example, $ f : a + b \sqrt{2} \to \sqrt{|a^2 - 2b^2|} $ does not define a norm on $ \mathbb Q(\sqrt{2}) $, as can be easily checked, for example since $ 2 \sqrt{2} = f(2 \sqrt{2}) = f((\sqrt{2} + 1) + f(\sqrt{2} - 1)) > f(\sqrt{2} + 1) + f(\sqrt{2} - 1) = 2 $, violating the triangle inequality. Your answer doesn't address this issue at all. | |
| Aug 24, 2021 at 20:58 | history | answered | Uri Bader | CC BY-SA 4.0 |