I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$:
\begin{alignat*}{2} p(X) &= (X-x_1)(X-x_2)\cdots(X-x_n) &=& \prod_{k=1}^n(X-x_k)\\ &= a_0 + a_1X + \cdots + a_{n-1}X^{n-1} + X^n &=& X^n+\sum_{k=0}^{n-1}a_kX^k \end{alignat*}
I'm interested in the polynomial
\begin{align*} r(Y) &= b_0 + b_1Y + \cdots + b_{n!-1}Y^{n!-1} + Y^{n!} = Y^{n!}+\sum_{k=0}^{n!-1}b_kY^k = \prod_{\pi\in S_n}\left(Y-\sum_{k=1}^n x_{\pi(k)}\omega^k\right) \\ &=\bigl(Y-x_1\omega-x_2\omega^2-\cdots-x_{n-1}\omega^n-x_n\bigr)\cdot \bigl(Y-x_2\omega-x_1\omega^2-\cdots\bigr)\cdots \end{align*}
where $\omega$ is a primitive $n$-th root of unity and $S_n$ is the symmeric group consisting of all permutations of $n$ elements.
I've read this formula in connection with the term “Lagrange resolvent”, but I'm not sure if that's really the correct term since Wikipedia uses that name only for the inner sum, without the outer product over all permutations. So if you know a better name for this beast, or know in more detail what Lagrange resolvent should refer to, please include that information in your answer or post a comment.
What I'm really trying to figure out, though, is how to express these $b_k$ from the corresponding $a_k$. It is obvious that doing so has to be possible, since the polynomial $r$ is obviously symmetric under $S_n$ and any symmeric polynomial can be expressed as a polynomial of the elementary symmetric functions, which are just the coefficients $a_k$ except for signs. So much for the theory. But how does one do this in practice?
I've worked out the first few examples, and unless I made a mistake, the relation is
- for $n=2$: \begin{align*}b_0&=a_1^2-4a_0\end{align*}
- for $n=3$: \begin{align*} b_0 &= a_2^6 - 9a_1a_2^4 + 27a_1^2a_2^2 - 27a_1^3 \\ b_3 &= 2a_2^3 - 9a_1a_2 + 27a_0 \end{align*}
- for $n=4$: this is a really long list, I'll paste that below.
All coefficients $b_k$ for $0<k<n!$ which I did not mention explicitely are equal to zero. Apparently coefficients are only non-zero for $k\equiv 0\pmod n$.
So my question is this: What's the most elegant way to compute these coefficients $b_k$ from the corresponding $a_k$? Doing so via the full symmetric polynomial in between seems unrealistic for $n\ge 6$ or so. Of course, even the list of all coefficients will become terribly large pretty soon, due to the factorial involved, but I'd hope that for $n\le 7$ there might be a more clever way to compute these $b_k$ coefficients from the $a_k$ in a practical way. How?
Here is the relation for $n=4$, as promised above:
\begin{align*} b_0 &= a_3^{24} - 32a_2a_3^{22} + 464a_2^2a_3^{20} + 16a_1a_3^{21} \\&\phantom= - 4032a_2^3a_3^{18} - 416a_1a_2a_3^{19} - 96a_0a_3^{20} + 23392a_2^4a_3^{16} \\&\phantom= + 4800a_1a_2^2a_3^{17} + 64a_1^2a_3^{18} + 2560a_0a_2a_3^{18} \\&\phantom= - 95488a_2^5a_3^{14} - 32384a_1a_2^3a_3^{15} - 1152a_1^2a_2a_3^{16} \\&\phantom= - 30336a_0a_2^2a_3^{16} - 1152a_0a_1a_3^{17} + 281344a_2^6a_3^{12} \\&\phantom= + 141568a_1a_2^4a_3^{13} + 8448a_1^2a_2^2a_3^{14} + 210432a_0a_2^3a_3^{14} \\&\phantom= - 128a_1^3a_3^{15} + 23808a_0a_1a_2a_3^{15} + 3456a_0^2a_3^{16} \\&\phantom= - 603136a_2^7a_3^{10} - 419328a_1a_2^5a_3^{11} - 31232a_1^2a_2^3a_3^{12} \\&\phantom= - 946688a_0a_2^4a_3^{12} + 3328a_1^3a_2a_3^{13} - 211968a_0a_1a_2^2a_3^{13} \\&\phantom= - 2304a_0a_1^2a_3^{14} - 73728a_0^2a_2a_3^{14} + 934144a_2^8a_3^8 \\&\phantom= + 852992a_1a_2^6a_3^9 + 52224a_1^2a_2^4a_3^{10} + 2887680a_0a_2^5a_3^{10} \\&\phantom= - 31744a_1^3a_2^2a_3^{11} + 1062912a_0a_1a_2^3a_3^{11} - 896a_1^4a_3^{12} \\&\phantom= + 24576a_0a_1^2a_2a_3^{12} + 678912a_0^2a_2^2a_3^{12} + 27648a_0^2a_1a_3^{13} \\&\phantom= - 1019904a_2^9a_3^6 - 1177600a_1a_2^7a_3^7 + 18432a_1^2a_2^5a_3^8 \\&\phantom= - 6051840a_0a_2^6a_3^8 + 153600a_1^3a_2^3a_3^9 - 3287040a_0a_1a_2^4a_3^9 \\&\phantom= + 10240a_1^4a_2a_3^{10} - 67584a_0a_1^2a_2^2a_3^{10} \\&\phantom= - 3526656a_0^2a_2^3a_3^{10} + 12288a_0a_1^3a_3^{11} \\&\phantom= - 423936a_0^2a_1a_2a_3^{11} - 55296a_0^3a_3^{12} + 745472a_2^{10}a_3^4 \\&\phantom= + 1056768a_1a_2^8a_3^5 - 249856a_1^2a_2^6a_3^6 + 8609792a_0a_2^7a_3^6 \\&\phantom= - 419840a_1^3a_2^4a_3^7 + 6426624a_0a_1a_2^5a_3^7 - 39936a_1^4a_2^2a_3^8 \\&\phantom= - 172032a_0a_1^2a_2^3a_3^8 + 11311104a_0^2a_2^4a_3^8 + 1024a_1^5a_3^9 \\&\phantom= - 180224a_0a_1^3a_2a_3^9 + 2666496a_0^2a_1a_2^2a_3^9 \\&\phantom= + 884736a_0^3a_2a_3^{10} - 327680a_2^{11}a_3^2 - 557056a_1a_2^9a_3^3 \\&\phantom= + 466944a_1^2a_2^7a_3^4 - 7962624a_0a_2^8a_3^4 + 659456a_1^3a_2^5a_3^5 \\&\phantom= - 7766016a_0a_1a_2^6a_3^5 + 53248a_1^4a_2^3a_3^6 \\&\phantom= + 1437696a_0a_1^2a_2^4a_3^6 - 22953984a_0^2a_2^5a_3^6 \\&\phantom= - 18432a_1^5a_2a_3^7 + 983040a_0a_1^3a_2^2a_3^7 \\&\phantom= - 8822784a_0^2a_1a_2^3a_3^7 + 18432a_0a_1^4a_3^8 \\&\phantom= + 221184a_0^2a_1^2a_2a_3^8 - 5824512a_0^3a_2^2a_3^8 \\&\phantom= - 221184a_0^3a_1a_3^9 + 65536a_2^{12} + 131072a_1a_2^{10}a_3 \\&\phantom= - 393216a_1^2a_2^8a_3^2 + 4325376a_0a_2^9a_3^2 - 557056a_1^3a_2^6a_3^3 \\&\phantom= + 5308416a_0a_1a_2^7a_3^3 + 34816a_1^4a_2^4a_3^4 \\&\phantom= - 3440640a_0a_1^2a_2^5a_3^4 + 28803072a_0^2a_2^6a_3^4 \\&\phantom= + 86016a_1^5a_2^2a_3^5 - 2555904a_0a_1^3a_2^3a_3^5 \\&\phantom= + 16220160a_0^2a_1a_2^4a_3^5 + 4096a_1^6a_3^6 - 49152a_0a_1^4a_2a_3^6 \\&\phantom= - 2064384a_0^2a_1^2a_2^2a_3^6 + 20217856a_0^3a_2^3a_3^6 \\&\phantom= - 221184a_0^2a_1^3a_3^7 + 2211840a_0^3a_1a_2a_3^7 + 331776a_0^4a_3^8 \\&\phantom= + 131072a_1^2a_2^9 - 1048576a_0a_2^{10} + 196608a_1^3a_2^7a_3 \\&\phantom= - 1572864a_0a_1a_2^8a_3 - 147456a_1^4a_2^5a_3^2 \\&\phantom= + 3735552a_0a_1^2a_2^6a_3^2 - 20447232a_0^2a_2^7a_3^2 \\&\phantom= - 155648a_1^5a_2^3a_3^3 + 3211264a_0a_1^3a_2^4a_3^3 \\&\phantom= - 15728640a_0^2a_1a_2^5a_3^3 - 8192a_1^6a_2a_3^4 \\&\phantom= - 221184a_0a_1^4a_2^2a_3^4 + 7176192a_0^2a_1^2a_2^3a_3^4 \\&\phantom= - 39059456a_0^3a_2^4a_3^4 - 73728a_0a_1^5a_3^5 \\&\phantom= + 1622016a_0^2a_1^3a_2a_3^5 - 8257536a_0^3a_1a_2^2a_3^5 \\&\phantom= + 442368a_0^3a_1^2a_3^6 - 3538944a_0^4a_2a_3^6 + 98304a_1^4a_2^6 \\&\phantom= - 1572864a_0a_1^2a_2^7 + 6291456a_0^2a_2^8 + 98304a_1^5a_2^4a_3 \\&\phantom= - 1572864a_0a_1^3a_2^5a_3 + 6291456a_0^2a_1a_2^6a_3 \\&\phantom= - 16384a_1^6a_2^2a_3^2 + 884736a_0a_1^4a_2^3a_3^2 \\&\phantom= - 11010048a_0^2a_1^2a_2^4a_3^2 + 39845888a_0^3a_2^5a_3^2 \\&\phantom= - 8192a_1^7a_3^3 + 344064a_0a_1^5a_2a_3^3 - 3932160a_0^2a_1^3a_2^2a_3^3 \\&\phantom= + 13631488a_0^3a_1a_2^3a_3^3 + 221184a_0^2a_1^4a_3^4 \\&\phantom= - 3538944a_0^3a_1^2a_2a_3^4 + 14155776a_0^4a_2^2a_3^4 + 32768a_1^6a_2^3 \\&\phantom= - 786432a_0a_1^4a_2^4 + 6291456a_0^2a_1^2a_2^5 - 16777216a_0^3a_2^6 \\&\phantom= + 16384a_1^7a_2a_3 - 393216a_0a_1^5a_2^2a_3 + 3145728a_0^2a_1^3a_2^3a_3 \\&\phantom= - 8388608a_0^3a_1a_2^4a_3 + 49152a_0a_1^6a_3^2 \\&\phantom= - 1179648a_0^2a_1^4a_2a_3^2 + 9437184a_0^3a_1^2a_2^2a_3^2 \\&\phantom= - 25165824a_0^4a_2^3a_3^2 + 4096a_1^8 - 131072a_0a_1^6a_2 \\&\phantom= + 1572864a_0^2a_1^4a_2^2 - 8388608a_0^3a_1^2a_2^3 + 16777216a_0^4a_2^4 \\ b_4 &= -6a_3^{20} + 160a_2a_3^{18} - 1872a_2^2a_3^{16} - 144a_1a_3^{17} \\&\phantom= + 12544a_2^3a_3^{14} + 3456a_1a_2a_3^{15} - 576a_0a_3^{16} - 52608a_2^4a_3^{12} \\&\phantom= - 35584a_1a_2^2a_3^{13} - 1536a_1^2a_3^{14} + 12288a_0a_2a_3^{14} \\&\phantom= + 141056a_2^5a_3^{10} + 204672a_1a_2^3a_3^{11} + 30336a_1^2a_2a_3^{12} \\&\phantom= - 113024a_0a_2^2a_3^{12} - 4992a_0a_1a_3^{13} - 233728a_2^6a_3^8 \\&\phantom= - 717568a_1a_2^4a_3^9 - 243968a_1^2a_2^2a_3^{10} + 581120a_0a_2^3a_3^{10} \\&\phantom= - 14976a_1^3a_3^{11} + 97536a_0a_1a_2a_3^{11} - 16512a_0^2a_3^{12} \\&\phantom= + 206848a_2^7a_3^6 + 1567744a_1a_2^5a_3^7 + 1029632a_1^2a_2^3a_3^8 \\&\phantom= - 1802240a_0a_2^4a_3^8 + 216832a_1^3a_2a_3^9 - 808960a_0a_1a_2^2a_3^9 \\&\phantom= + 8448a_0a_1^2a_3^{10} + 264192a_0^2a_2a_3^{10} - 33280a_2^8a_3^4 \\&\phantom= - 2082816a_1a_2^6a_3^5 - 2443264a_1^2a_2^4a_3^6 + 3366912a_0a_2^5a_3^6 \\&\phantom= - 1202176a_1^3a_2^2a_3^7 + 3584000a_0a_1a_2^3a_3^7 - 81024a_1^4a_3^8 \\&\phantom= + 55296a_0a_1^2a_2a_3^8 - 1677312a_0^2a_2^2a_3^8 - 251904a_0^2a_1a_3^9 \\&\phantom= - 94208a_2^9a_3^2 + 1538048a_1a_2^7a_3^3 + 3217408a_1^2a_2^5a_3^4 \\&\phantom= - 3512320a_0a_2^6a_3^4 + 3194880a_1^3a_2^3a_3^5 \\&\phantom= - 8792064a_0a_1a_2^4a_3^5 + 788480a_1^4a_2a_3^6 \\&\phantom= - 1400832a_0a_1^2a_2^2a_3^6 + 5378048a_0^2a_2^3a_3^6 \\&\phantom= + 227328a_0a_1^3a_3^7 + 2635776a_0^2a_1a_2a_3^7 + 387072a_0^3a_3^8 \\&\phantom= + 57344a_2^{10} - 483328a_1a_2^8a_3 - 2117632a_1^2a_2^6a_3^2 \\&\phantom= + 1613824a_0a_2^7a_3^2 - 4085760a_1^3a_2^4a_3^3 \\&\phantom= + 11153408a_0a_1a_2^5a_3^3 - 2699264a_1^4a_2^2a_3^4 \\&\phantom= + 6725632a_0a_1^2a_2^3a_3^4 - 9181184a_0^2a_2^4a_3^4 - 139264a_1^5a_3^5 \\&\phantom= - 1335296a_0a_1^3a_2a_3^5 - 9449472a_0^2a_1a_2^2a_3^5 \\&\phantom= - 1179648a_0^2a_1^2a_3^6 - 4128768a_0^3a_2a_3^6 + 499712a_1^2a_2^7 \\&\phantom= - 65536a_0a_2^8 + 2019328a_1^3a_2^5a_3 - 5668864a_0a_1a_2^6a_3 \\&\phantom= + 3837952a_1^4a_2^3a_3^2 - 11964416a_0a_1^2a_2^4a_3^2 \\&\phantom= + 8028160a_0^2a_2^5a_3^2 + 706560a_1^5a_2a_3^3 \\&\phantom= + 1581056a_0a_1^3a_2^2a_3^3 + 13238272a_0^2a_1a_2^3a_3^3 \\&\phantom= + 665600a_0a_1^4a_3^4 + 5947392a_0^2a_1^2a_2a_3^4 \\&\phantom= + 15351808a_0^3a_2^2a_3^4 + 3489792a_0^3a_1a_3^5 - 1898496a_1^4a_2^4 \\&\phantom= + 7110656a_0a_1^2a_2^5 - 2883584a_0^2a_2^6 - 815104a_1^5a_2^2a_3 \\&\phantom= + 229376a_0a_1^3a_2^3a_3 - 6160384a_0^2a_1a_2^4a_3 - 159744a_1^6a_3^2 \\&\phantom= - 1671168a_0a_1^4a_2a_3^2 - 3833856a_0^2a_1^2a_2^2a_3^2 \\&\phantom= - 23199744a_0^3a_2^3a_3^2 - 2818048a_0^2a_1^3a_3^3 \\&\phantom= - 18415616a_0^3a_1a_2a_3^3 - 3637248a_0^4a_3^4 + 212992a_1^6a_2 \\&\phantom= + 704512a_0a_1^4a_2^2 - 3276800a_0^2a_1^2a_2^3 + 12582912a_0^3a_2^4 \\&\phantom= + 638976a_0a_1^5a_3 + 1376256a_0^2a_1^3a_2a_3 + 17301504a_0^3a_1a_2^2a_3 \\&\phantom= + 9895936a_0^3a_1^2a_3^2 + 19398656a_0^4a_2a_3^2 - 1277952a_0^2a_1^4 \\&\phantom= + 1572864a_0^3a_1^2a_2 - 18874368a_0^4a_2^2 - 20971520a_0^4a_1a_3 \\&\phantom= + 16777216a_0^5 \\ b_8 &= 15a_3^{16} - 320a_2a_3^{14} + 2848a_2^2a_3^{12} + 416a_1a_3^{13} \\&\phantom= - 13440a_2^3a_3^{10} - 7872a_1a_2a_3^{11} + 1984a_0a_3^{12} + 34880a_2^4a_3^8 \\&\phantom= + 59520a_1a_2^2a_3^9 + 5248a_1^2a_3^{10} - 31744a_0a_2a_3^{10} \\&\phantom= - 44800a_2^5a_3^6 - 222080a_1a_2^3a_3^7 - 85120a_1^2a_2a_3^8 \\&\phantom= + 196480a_0a_2^2a_3^8 + 45440a_0a_1a_3^9 + 16640a_2^6a_3^4 \\&\phantom= + 387840a_1a_2^4a_3^5 + 536320a_1^2a_2^2a_3^6 - 581120a_0a_2^3a_3^6 \\&\phantom= + 5760a_1^3a_3^7 - 528640a_0a_1a_2a_3^7 - 24960a_0^2a_3^8 \\&\phantom= + 3072a_2^7a_3^2 - 165376a_1a_2^5a_3^3 - 1523200a_1^2a_2^3a_3^4 \\&\phantom= + 816640a_0a_2^4a_3^4 - 206080a_1^3a_2a_3^5 + 2073600a_0a_1a_2^2a_3^5 \\&\phantom= + 456960a_0a_1^2a_3^6 + 266240a_0^2a_2a_3^6 + 16640a_2^8 \\&\phantom= - 211968a_1a_2^6a_3 + 1571840a_1^2a_2^4a_3^2 - 487424a_0a_2^5a_3^2 \\&\phantom= + 1397760a_1^3a_2^2a_3^3 - 2877440a_0a_1a_2^3a_3^3 - 105600a_1^4a_3^4 \\&\phantom= - 3420160a_0a_1^2a_2a_3^4 - 947200a_0^2a_2^2a_3^4 - 353280a_0^2a_1a_3^5 \\&\phantom= + 172032a_1^2a_2^5 + 159744a_0a_2^6 - 2611200a_1^3a_2^3a_3 \\&\phantom= + 512000a_0a_1a_2^4a_3 + 20480a_1^4a_2a_3^2 + 6563840a_0a_1^2a_2^2a_3^2 \\&\phantom= + 1064960a_0^2a_2^3a_3^2 + 1628160a_0a_1^3a_3^3 \\&\phantom= + 2785280a_0^2a_1a_2a_3^3 - 430080a_0^3a_3^4 + 1387520a_1^4a_2^2 \\&\phantom= - 655360a_0a_1^2a_2^3 + 286720a_0^2a_2^4 - 271360a_1^5a_3 \\&\phantom= - 5775360a_0a_1^3a_2a_3 - 5980160a_0^2a_1a_2^2a_3 \\&\phantom= - 1105920a_0^2a_1^2a_3^2 + 2293760a_0^3a_2a_3^2 + 1085440a_0a_1^4 \\&\phantom= + 7208960a_0^2a_1^2a_2 - 1638400a_0^3a_2^2 - 4259840a_0^3a_1a_3 \\&\phantom= + 4259840a_0^4 \\ b_{12} &= -20a_3^{12} + 320a_2a_3^{10} - 1952a_2^2a_3^8 - 544a_1a_3^9 \\&\phantom= + 5376a_2^3a_3^6 + 7040a_1a_2a_3^7 - 1536a_0a_3^8 - 5504a_2^4a_3^4 \\&\phantom= - 31488a_1a_2^2a_3^5 - 6144a_1^2a_3^6 + 16384a_0a_2a_3^6 - 768a_2^5a_3^2 \\&\phantom= + 49792a_1a_2^3a_3^3 + 59264a_1^2a_2a_3^4 - 55424a_0a_2^2a_3^4 \\&\phantom= - 30336a_0a_1a_3^5 + 1280a_2^6 - 8448a_1a_2^4a_3 - 136960a_1^2a_2^2a_3^2 \\&\phantom= + 51712a_0a_2^3a_3^2 - 56192a_1^3a_3^3 + 210688a_0a_1a_2a_3^3 \\&\phantom= - 12672a_0^2a_3^4 + 512a_1^2a_2^3 + 31744a_0a_2^4 + 248064a_1^3a_2a_3 \\&\phantom= - 397312a_0a_1a_2^2a_3 - 69888a_0a_1^2a_3^2 + 67584a_0^2a_2a_3^2 \\&\phantom= - 172928a_1^4 + 391168a_0a_1^2a_2 + 12288a_0^2a_2^2 - 307200a_0^2a_1a_3 \\&\phantom= + 409600a_0^3 \\ b_{16} &= 15a_3^8 - 160a_2a_3^6 + 528a_2^2a_3^4 + 336a_1a_3^5 \\&\phantom= - 448a_2^3a_3^2 - 2208a_1a_2a_3^3 - 96a_0a_3^4 - 160a_2^4 \\&\phantom= + 2752a_1a_2^2a_3 + 2368a_1^2a_3^2 + 512a_0a_2a_3^2 - 3328a_1^2a_2 \\&\phantom= + 2304a_0a_2^2 - 8960a_0a_1a_3 + 17920a_0^2 \\ b_{20} &= -6a_3^4 + 32a_2a_3^2 - 16a_2^2 - 80a_1a_3 + 320a_0 \end{align*}