You might look at p-adic numbers. For example, sqrt(2) mod 7^N is (radix 7) +-...13 . Although some folks are uncomfortable about 2 being closer to 345 than to 9 or 51. I think you can get some milage mapping sqrt(6) (mod 7^N) to an "imaginary" 7-adic, adjoining i = sqrt(-1). You could also adjoin (rational) fractional powers to allow sqrt(7). I have no clue whether this is a complete representation system for algebraics (mod 7^N). The three cube roots of 1 seem straightforward, with sevens&units digits 01, 42, and 24. cbrt(2) (mod 7^N) looks like a problem. Even if we extend by all J^(1/K), we still face nested radicals, and non-radical algebraics. An interesting puzzle to see how far this can be pushed. Rich ----------- Quoting Henry Baker <hbaker1@pipeline.com>:
Yes, you are technically correct.
However, I was thinking about the naive freshman engineering class where we plot roots on the complex plane to determine -- e.g., the stability of some linear system. It's a lot shorter conversation if we plot them in the usual (uncountable) complex plane.
Now that finite fields are so important in crypto and coding theory, where are these freshmen (& bright high schoolers) going to find their roots?
At 08:31 AM 7/22/2017, Eugene Salamin via math-fun wrote:
The algebraic closure of the rationals consists of the algebraic numbers, and is a countable field.
The complex numbers are an uncountable field.
In between the two lie the algebraic closures of the various transcendental extensions of the rationals.
-- Gene
On Friday, July 21, 2017, 5:46:55 PM PDT, Henry Baker <hbaker1@pipeline.com> wrote:
Thanks, Dan.
The algebraic closure of the infinite sequence GF(p),GF(p^2),GF(p^3),... as described in the Wikipedia article is pretty ugly -- at least as described.
I was hoping for something as elegant as the complex numbers with their usual topology.
Perhaps I need to check back in about 300 years?
At 10:08 AM 7/21/2017, Dan Asimov wrote:
Every field K -- if it does not alreeady contain all the roots of polynomials in K[x] -- iss a proper subfield of a larger field that is algebraically closed.
E.g., see <https://en.wikipedia.org/wiki/Finite_field#Algebraic_closure>.
I don't know how such an algebraic closure is usually visualized.
But I believe that, for any prime p, the Galois group of [the algebraic closure of the finite field F(p)] over F(p) is isomorphic to the same thing for any other prime.
--Dan
From: Henry Baker <hbaker1@pipelline.com> Jul 21, 2017 9:06 AM: -----
OK, if I extend the rationals with the root alpha of an irreducible polynomial p[x], I can plot alpha on the complex plane; indeed, I can plot *all* of the roots of p[x] on the complex plane. So all of these "extension roots" live in the complex plane.
Is there an analogous (single) place/field where all extension roots of GF(p) live -- i.e., a larger field which includes all of the extension fields of GF(p) ?
There seems to be a problem, since there are many (isomorphic) ways to extend GF(p); perhaps these are all different in this larger field?
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