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Hi. I'm working through a p-adic analysis book, and there's a quick review of fields at the beginning of chapter 3. I read the following there:
A field extension K of F is an F-vector space; if it is finite-dimensional, it must be an algebraic extension, and its dimension is called the degree [K:F]. If has the property that every element of K can be written as a rational expression in , we write and say that K is the extension obtained by adjoining to F.
Two paragraphs later:
If F is a perfect field, then any finite extension K of F is of the form for some . is called a primitive element. Knowing a primitive element of a field extension K makes it easier to study K, since it means that everything in K is a polynomial in of degree <n.
Super. So... the first paragraph there says that everything in K is a rational expression in . The second says that everything in K is a polynomial in . Those aren't equivalent, are they? What polynomial in is the same as , for example? What am I missing? Is the polynomial thing something extra that we get when F is perfect? - GTBacchus( talk) 00:10, 19 September 2009 (UTC)
I learned one porcelain bowl from the estate of Barbara Hutton was auctioned and sold by Christie's Hong Kong for $22,240,000 in 2006. When I tried to use the inflation calculator provided by the United States Department of Labor to figure out what that sum of money would be today, I ran into a problem. The original sum has to be at least $10,000,000 or less. So if anyone could improvise and help me figure out what $22,240,000 in 2006 would be in today's money, that would be great. Thank you. 69.203.157.50 ( talk) 05:43, 19 September 2009 (UTC)
I am interested in algorithmically generating the symmetries of a 2D or 3D point lattice, given reduced lattice basis vectors. I know that there are a finite number of crystallographic point groups in these dimensions, but I am wondering how those tables of symmetries were generated. Is there an automatic way of discovering lattice symmetries? Victor Liu ( talk) 19:39, 19 September 2009 (UTC)
Mathematics desk | ||
---|---|---|
< September 18 | << Aug | September | Oct >> | September 20 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Hi. I'm working through a p-adic analysis book, and there's a quick review of fields at the beginning of chapter 3. I read the following there:
A field extension K of F is an F-vector space; if it is finite-dimensional, it must be an algebraic extension, and its dimension is called the degree [K:F]. If has the property that every element of K can be written as a rational expression in , we write and say that K is the extension obtained by adjoining to F.
Two paragraphs later:
If F is a perfect field, then any finite extension K of F is of the form for some . is called a primitive element. Knowing a primitive element of a field extension K makes it easier to study K, since it means that everything in K is a polynomial in of degree <n.
Super. So... the first paragraph there says that everything in K is a rational expression in . The second says that everything in K is a polynomial in . Those aren't equivalent, are they? What polynomial in is the same as , for example? What am I missing? Is the polynomial thing something extra that we get when F is perfect? - GTBacchus( talk) 00:10, 19 September 2009 (UTC)
I learned one porcelain bowl from the estate of Barbara Hutton was auctioned and sold by Christie's Hong Kong for $22,240,000 in 2006. When I tried to use the inflation calculator provided by the United States Department of Labor to figure out what that sum of money would be today, I ran into a problem. The original sum has to be at least $10,000,000 or less. So if anyone could improvise and help me figure out what $22,240,000 in 2006 would be in today's money, that would be great. Thank you. 69.203.157.50 ( talk) 05:43, 19 September 2009 (UTC)
I am interested in algorithmically generating the symmetries of a 2D or 3D point lattice, given reduced lattice basis vectors. I know that there are a finite number of crystallographic point groups in these dimensions, but I am wondering how those tables of symmetries were generated. Is there an automatic way of discovering lattice symmetries? Victor Liu ( talk) 19:39, 19 September 2009 (UTC)