quickfur wrote:First step is to derive algebraic representations of sin(pi/7) and cos(pi/7), for obvious reasons. There are several approaches for this; the approach I adopted is to start with 0 = sin(pi) = sin(7*pi/7) = sin(8*pi/7 - pi/7). This then can be split using the angle addition trig identity into sines and cosines involving 8*pi/7 and pi/7, the former of which can be successively reduced to pi/7 using the double angle formulas. At first, I set about doing this by hand, but the expressions quickly got far too unwieldy to handle, so I decided on a different approach.
quickfur wrote:Now, here's an interesting point. I tried to derive a single defining polynomial for H, but it turned out that the minimal such polynomial is a whopping degree-36 polynomial with gargantuan coefficients (about 40 digits or so per coefficient). Yeah, you read that right, the leading term is x36. It only has even powers, though, so if you take it as a polynomial in x2 it would be an 18th degree polynomial -- but that's still huge (and the coefficients remain gigantic 40+-digit numbers). And this nasty thing is irreducible. What's interesting about this is that h and r, by themselves, have very tame-looking polynomials. Yet their sum caused an explosion in polynomial degree and magnitude of coefficients. This makes me wonder if there any known algorithm or method for splitting an algebraic number of high degree into a sum of two algebraic numbers of significantly smaller degree, or at least smaller coefficients? Currently, the best polynomials I've found for, e.g., J88 are degree 16, with moderate-sized coefficients, and for J89 the polynomials are degree 12 with largish coefficients. It would be nice to find much smaller polynomials whose roots, when summed together, produce the same value. Is there any literature out there on this subject?
mr_e_man wrote:quickfur wrote:[...]I tried to derive a single defining polynomial for H, but it turned out that the minimal such polynomial is a whopping degree-36 polynomial with gargantuan coefficients (about 40 digits or so per coefficient). Yeah, you read that right, the leading term is x^36. It only has even powers, though, so if you take it as a polynomial in x2 it would be an 18th degree polynomial -- but that's still huge (and the coefficients remain gigantic 40+-digit numbers). And this nasty thing is irreducible. What's interesting about this is that h and r, by themselves, have very tame-looking polynomials. Yet their sum caused an explosion in polynomial degree and magnitude of coefficients. This makes me wonder if there any known algorithm or method for splitting an algebraic number of high degree into a sum of two algebraic numbers of significantly smaller degree, or at least smaller coefficients? [...]
This question appears on MSE, though the answers look unsatisfactory.
In general, if x and y are algebraic numbers of degree m and n respectively (so the fields ℚ(x) and ℚ(y) have dimensions m and n), then any polynomial combination of them such as x+y or x*y has degree at most m*n, because the field ℚ(x,y) has dimension m*n or smaller.
[...]
I found here a possible method for decomposing an algebraic number α, by finding sub-fields of ℚ(α). Take its minimal polynomial (which doesn't factor over ℚ), factor it into smaller polynomials with coefficients in ℚ(α), look at the sub-fields generated by the coefficients in each factor, and try to write α as a sum of numbers from two different sub-fields. I don't know if this is helpful.
quickfur wrote:Also, are there any bounds on the coefficients, or they can explode without bound when adding two arbitrary algebraic numbers?
quickfur wrote:The theorem sounds promising, but factoring the minimal polynomial will not help, because since α is a root of its minimal polynomial, its minimal polynomial in ℚ(α) is nothing but x - α. What you want is to find some intermediate field L between ℚ and ℚ(α), which according to your link above, ought to have a degree that divides the degree of ℚ(α), that lies along the extension chain from ℚ to ℚ(α). If you can find such a field, then you could meaningfully talk about factoring the minimal polynomial of α over L.
mr_e_man wrote:[...]quickfur wrote:Also, are there any bounds on the coefficients, or they can explode without bound when adding two arbitrary algebraic numbers?
And polynomial functions are bounded by powers of bounds on the inputs; for example, if a,b,c,d are integers, and we write m=max(|a|,|b|,|c|,|d|), then
|a³+2ab+6ad+4cd| ≤ |a|³+2|a||b|+6|a||d|+4|c||d| ≤ m³+2m²+6m²+4m² = m³+12m² ≤ m³+12m³ = 13m³.
quickfur wrote:The theorem sounds promising, but factoring the minimal polynomial will not help, because since α is a root of its minimal polynomial, its minimal polynomial in ℚ(α) is nothing but x - α. What you want is to find some intermediate field L between ℚ and ℚ(α), which according to your link above, ought to have a degree that divides the degree of ℚ(α), that lies along the extension chain from ℚ to ℚ(α). If you can find such a field, then you could meaningfully talk about factoring the minimal polynomial of α over L.
No, I meant to take the minimal polynomial over ℚ, and factor that over ℚ(α). Say α is the 6th root of 2; its minimal polynomial is x⁶ - 2, which factors as
x⁶ - 2 = (x³ - √2) (x³ + √2) = (x³ - α³) (x³ + α³)
[...]
mr_e_man wrote:[...]
Where did that 36th-degree thing come from?
7*C^6 - 147*C^4 + 917*C^2 - 1681
C^6 - 21*C^4 + 35*C^2 - 7
7*C^6 - 35*C^4 + 21*C^2 - 1
7*C^6 - 91*C^4 + 245*C^2 - 169
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