We had the following problem for homework: let prove that lies in a cyclotomic extension.
(Without using Kronecker-Weber Theorem!)

I found a solution to this problem using quadradic Gauss sums, however this is not a course in number theory - it is a course on Galois theory, therefore my solution might be not acceptable. I will post it anyway because I want to print it out and give it to the class so which is why I am hanging it on the forum. I am interested in seeing other solutions.

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

Hi

Here is your PDF file. Let me know if anything has to be modified.

Attached Files

tph.pdf (68.9 KB, 29 views)

Thank you very much!

There are two mistakes that I did. The first mistake appears in the proof of Theorem 1. The part in the parenthesis "If this fact is not familar ... "
After Theorem 3, "As a result ... ", that paragraph. It should say and not .

I edited my post above if it makes it any easier.

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

without your respect sir, can i ask why you can't do pdf?
i'm just curious.. Ü hope, you won't mind..

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I can do a PDF, but why should I do it if I can find someone else to do it?
Just joking! I never made a PDF before, I do not have a program which allows me to make a PDF, and I do not have LaTeX installed on my computer.

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

hahaha.. ok.

if you are interested in installing one, try downloading it here..

* > > > OVERJOYED < < < - LaTex.. LaTex.. LaTex

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I've corrected these two mistakes, see attachment.

Attached Files

tph_2.pdf (69.0 KB, 28 views)

You may want to increase the size of the document or modify layout options.

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• Got trouble with LaTeX? Take a look.
  
• Make your partial integration (aka integration by parts) faster. Click here.
  
• How to prove limits by the definition.

Thank you very much, you made it look like a professional written paper.
I am sorry that France lost yesterday.

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

Well that's why it's interesting to use LaTeX : once one knows the commands used to write equations (like \frac{}{} ...) it's really easy to make a document like this one.
It's been a while since I've watched a football match on TV so I won't comment on this.