[math-fun] Matrices to represent extension field elements
A recent post mentioned that (square) matrices were a kind of "universal" representation for fields, in that a matrix A whose characteristic polynomial is p[x] can represent a root of this polynomial in the extension field. The problem is that such a matrix doesn't represent a *single* root of p[x], but simultaneously represents *all* of the roots of p[x]. (Hint: diagonalize the matrix so that the main diagonal consists of the eigenvalues, i.e., all the roots of p[x].) Suppose we have the simplest situation: a symmetric matrix with all distinct real roots. Since the matrix itself represents *all* of these roots simultaneously, it can't participate, e.g., in ordering comparisons with other numbers which may lie in the middle of this set of roots. So there needs to be some way to distinguish exactly which root we're talking about when given the matrix A, before we can realistically utilize this matrix as a representation of one of the real roots of the polynomial. --- In the usual case of real matrices with complex eigenvalues, the coefficients of the polynomial, the entries of the matrix, and the roots themselves are usually more-or-less continuous functions of one another. So if we vary a polynomial coefficient by a little, one or more of the roots will vary in a more-or-less predictable way. If we're going to extend finite fields in a p-adic way, then presumably there will be an analogous "continuity" between the polynomial coefficients and the roots. It is my understanding that all of the fields GF(p^k) with a given prime p and a given positive power k are isomorphic to one another. So does this mean that instead of extending by p^k, one could instead extend first by p^i, and subsequently by p^j, such that i+j=k ? Perhaps one could build (always?) some sort of "tower" of extensions whereby each field is embedded in the next ? Under these conditions, could the polynomial coefficients be "close" to those of the extensions, and the polynomial roots be "close" to those of the extension, where "close" is measured in some p-adic way?
Addressing only the third-to-last paragraph, about extensions & towers: If j and k are relatively prime positive numbers, you can develop the fields GF[p^j], GF[p^k], and GF[p^(jk)]. The last contains the first two as subfields. If you select irreducible polynomials (mod p) of degrees j and k, you can adjoin roots of both polynomials to GF[p] and get the product degree field GF[p^(jk)]. It works, I've written the code, it runs. If j divides k, then GF[p^j] is embeddable as a subfield of GF[p^k]; if you've selected a representation of GF[p^k], there's a unique subset that makes up GF[p^j]. The embedding is unique as to the set of elements, but not wrt the actual mapping, because of automorphisms of GF[p^j]. If you choose an N, and consider its divisibility box, with 1 at the lower left front corner, and N at the opposite corner, each node J of the box has a field GF[p^J], and the subfields have the same relationship as the box nodes. Rich -------- Quoting hbaker1 <hbaker1@pipeline.com>:
A recent post mentioned that (square) matrices were a kind of "universal" representation for fields, in that a matrix A whose characteristic polynomial is p[x] can represent a root of this polynomial in the extension field.
The problem is that such a matrix doesn't represent a *single* root of p[x], but simultaneously represents *all* of the roots of p[x]. (Hint: diagonalize the matrix so that the main diagonal consists of the eigenvalues, i.e., all the roots of p[x].)
Suppose we have the simplest situation: a symmetric matrix with all distinct real roots. Since the matrix itself represents *all* of these roots simultaneously, it can't participate, e.g., in ordering comparisons with other numbers which may lie in the middle of this set of roots.
So there needs to be some way to distinguish exactly which root we're talking about when given the matrix A, before we can realistically utilize this matrix as a representation of one of the real roots of the polynomial.
---
In the usual case of real matrices with complex eigenvalues, the coefficients of the polynomial, the entries of the matrix, and the roots themselves are usually more-or-less continuous functions of one another. So if we vary a polynomial coefficient by a little, one or more of the roots will vary in a more-or-less predictable way.
If we're going to extend finite fields in a p-adic way, then presumably there will be an analogous "continuity" between the polynomial coefficients and the roots.
It is my understanding that all of the fields GF(p^k) with a given prime p and a given positive power k are isomorphic to one another. So does this mean that instead of extending by p^k, one could instead extend first by p^i, and subsequently by p^j, such that i+j=k ?
Perhaps one could build (always?) some sort of "tower" of extensions whereby each field is embedded in the next ?
Under these conditions, could the polynomial coefficients be "close" to those of the extensions, and the polynomial roots be "close" to those of the extension, where "close" is measured in some p-adic way?
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