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In mathematics and, more specifically, in theory of equations, the principal form of an irreducible polynomial of degree at least three is a polynomial of the same degree n without terms of degrees n−1 and n−2, such that each roots of either polynomial is a rational function of a root of the other polynomial.

The principal form of a polynomial can be found by applying a suitable Tschirnhaus transformation to the given polynomial.

Definition

Let

${\displaystyle f(x)=x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}}$

be an irreducible polynomial of degree at least three.

Its principal form is a polynomial

${\displaystyle g(y)=y^{n}+b_{3}y^{n-3}+\cdots +b_{n-1}y+b_{n},}$

together with a Tschirnhaus transformation of degree two

${\displaystyle \varphi (x)=x^{2}+\alpha x+\beta }$

such that, if r is a root of f, ${\displaystyle \phi (r)}$ is a root of ⁠${\displaystyle g}$⁠.

Expressing that ⁠${\displaystyle g}$⁠ does not has terms in ⁠${\displaystyle y^{n-1}}$⁠ and ⁠${\displaystyle y^{n-2}}$⁠ leads to a system of two equations in ⁠${\displaystyle \alpha }$⁠ and ⁠${\displaystyle \beta }$⁠, one of degree one and one of degree two. In general, this system has two solutions, giving two principal forms involving a square root. One passes from one principal form to the secong by changing the sign of the square root.

Literature

• "Polynomial Transformations of Tschirnhaus", Bring and Jerrard, ACM Sigsam Bulletin, Vol 37, No. 3, September 2003
• F. Brioschi, Sulla risoluzione delle equazioni del quinto grado: Hermite — Sur la résolution de l'Équation du cinquiéme degré Comptes rendus —. N. 11. Mars. 1858. 1. Dezember 1858, doi:10.1007/bf03197334
• Bruce and King, Beyond the Quartic Equation, Birkhäuser, 1996.

References

1. Weisstein, Eric W. "Principal Quintic Form". mathworld.wolfram.com.
2. "The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$". Mathematics Stack Exchange.
3. Jerrard, George Birch (1859). An essay on the resolution of equations. London, UK: Taylor & Francis.
4. Adamchik, Victor (2003). "Polynomial Transformations of Tschirnhaus, Bring, and Jerrard" (PDF). ACM SIGSAM Bulletin. 37 (3): 91. CiteSeerX 10.1.1.10.9463. doi:10.1145/990353.990371. S2CID 53229404. Archived from the original (PDF) on 2009-02-26.

• Algebra
• Equations
• Galois theory
• Polynomials
• Polynomial functions
• Nonlinear systems