For finite fields, every element can be written as a finite power of a generator, and there are typically two different bases used: - the polynomial basis, where elements are written as sums of consecutive powers of a generator - the normal basis, where elements are written as sums of (prime-power) powers of an element For example, take polynomials over F_2 mod x^4-x-1. Then x is a generator and we can write every element y in the polynomial basis as y = \sum_{i=0}^3 y_i x^i where y_i is in F_2. Let z=x^3. Then we can also write y in the normal basis as y = \sum_{j=1}^4 y_j z^{2^j} Let a_n = lcm(1,...,n). Then F_{p^{a_n}} contains a copy of F_{p^{a_m}} for each m less than n. "The" limit (there are actually several, but they're all isomorphic as fields) is the algebraic closure of F_p. I haven't been able to find any information about how to write down specific elements of this field. It looks like there's an "infinitesimal" generator x, and every element of the field can be written as a rational root of unity times a finite power of x. Is that right? Are there bases corresponding to the polynomial or normal basis for the closure? If not, what kind of bases are there? -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike