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Of course not every tensor product is a field extension, but what about the converse? The multiplication of dimensions makes me suspect this, but it's been awhile since I "studied" this stuff.(but i did ask a teacher if it was a tensor product and he said yes, on the other hand, he didn't spend any
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I think that having the definition of a subfield contained in this article on field extensions is a little odd. Moreover, the search for subfield (mathematics) directs to this field extension page. The definition of subfield should be in the field article and then, given proper links, knowledge of
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Close. The vector space is |K|^(n), the direct sum of n copies of K, not the direct product. In this case Q^(aleph0) has cardinality aleph0, the cardinality of all finite sequences (of unbounded length) of rational numbers. The direct product Q^aleph0 has cardinality c, the cardinality of all
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It is stated that the degree of K(t) over K, for any field K, has countable degree (N_0). I am certainly no expert, but this seems to be false. Any transcendental extension (i.e. K(t)) must have degree at last the cardinality of K, since all the inverses of (1 - x t), for x in K, are linearly
321:, and to flesh out that article a bit, including a definition of degree (with the infinite case as well), and perhaps a proof of the multiplicitivity result, and some examples. Also, I don't remember "degree theorem" being a standard term for this result. What do you think?
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I think I remember from math class that it's unknown whether every finite-degree extension is a Galois extension. If that's the case, it should be mentioned here. That always seemed like an amazingly basic open question, if it is one.
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OK, well that was probably more than I intended to write in one go :-) But now we have some overlap, which is not necessarily a bad thing, but perhaps we could focus the examples a bit better.
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time pondering his yes. Also, a different teacher denied that it was a tensor product.) If I'm right, some words on that and a link to tensor product would be valuable.
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could be redistributed in a better way. I would also like to see an example of a galois extension which is not abelian, if a simple example exists.
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I am currently restructering the article in order to collect the important definitions, which were scattered in the article, in the
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The question you half-remembered from math class is this: is every finite group the Galois group of some Galois extension of
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Sounds good, I like the new title. But I cannot help you fleshing out the article as I only know about the finite case.
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Just a sidenote, perhaps soemone can clean up the symbols in this page? It's kind of distracting to read...
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has finite degree, is normal, but it is not separable, so it is not a Galois extension. --
686:{\displaystyle |\mathbb {Q} |^{\aleph _{0}}=\aleph _{0}^{\aleph _{0}}=2^{\aleph _{0}}=c}
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If you wish to start a new discussion or revive an old one, please do so on the
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and not directly relevant to the article. I added a link to the theorem in the
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is a relevant search term, though I forget if wikipedia has much on it.
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countable sequence of rational numbers. So really is c. I think
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The statement was wrong, so I've removed it from the article. --
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Umm, it's the other way around. Every algebraic extension of
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589:{\displaystyle \aleph _{0}}
319:degree of a field extension
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301:19:38, 17 April 2006 (UTC)
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267:13:32, 17 April 2006 (UTC)
244:07:15, 23 March 2006 (UTC)
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147:23:38, 13 March 2006 (UTC)
94:00:24, 13 March 2006 (UTC)
194:17:50, 15 July 2005 (UTC)
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790:Too Compact
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100:merge with
36:This is an
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296:section.
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