Newton's Identities

In 1707, Sir Isaac Newton published a set of identities that would later become very important in the establishment of group theory. He showed an important relationship between the roots of an equation and the coefficients that make up the equation:

Let r,s,t be the roots of a cubic equation x³ + bx² + cx + d=0 (See here for details on the cubic equation and the proof that it always has three solutions).

Then, the following are true:

(1) r + s + t = -b

(2) r² + s² + t² = b² - 2c

(3) r³ + s³ + t³ = -b³ + 3bc - 3d

(4) rs + rt + st = c

(5) r²s + r²t + s²r + s²t + t²r + t²s = -bc + 3d

(6) r³s + r³t + s³r + s³t + t³r + t³s = b²c - 2c² - bd

(7) r²s² + r²t² + s²t² = c² - 2bd

(8) r³s² + r³t² + s³r² + s³t² + t³r² + t³s² = -bc² + 2b²d + cd

(9) r³s³ + r³t³ + s³t³ = c³ - 3bcd + 3d²

(10) rst = -d

(11) r²st + s²rt + t²rs = bd

(12) r³st + s³rt + t³rs = -b²d + 2cd

(13) r²s²t + r²st² + rs²t² = -cd

(14) r³s²t + r³st² + s³r²t + s³t²r + t³r²s + t³s²r = bcd - 3d²

(15) r³s³t + r³t³s + s³r³t + s³t³r + t³r³s + t³s³r = -c²d + 2bd²

(16) r²s²t² = d²

(17) r³s²t² + s³r²t² + t³r²s² = -bd²

(18) r³s³t² + r³t³s² + s³t³r² = cd²

(19) r³s³t³ = -d³

By this time, Newton was already world famous and he did not provide any proof or method for coming up with these identities. Apparently, he did not know that Albert Girard had independently come up with many of the same identities also without offering proof.

How did he come up with these identities?

Apparently, Newton was trying to figure out a method to determine when two cubic equations had the same root. Later, from this investigation, he found these identities. Even though these identities are specific to the cubic equation, it turns out that underlying these identities is a general theorem that applies to any polynomial of any power. Harold Edwards calls this general theorem "Newton's Theorem."

In my next blog, I will show how these identities can be derived.

References

• Harold M. Edwards, Galois Theory, Springer, 1984.