Its principal form is a polynomial together with a Tschirnhaus transformation of degree two such that, if r is a root of f,
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.
To fulfill these criteria a separate equation system of several unknowns has to be solved.
The quadratic radical components[6] of the coefficients are identical to the square root terms appearing along with the Cardano theorem and therefore the Cubic Tschirnhaus transformation even can be used to derive the general Cardano formula itself.
Plastic constant: Supergolden constant: Tribonacci constant: The direct solving of the mentioned system of three clues leads to the Cardano formula for the mentioned case: This is the given quartic equation: Now this quadratic equation system shall be solved: And so accurately that Tschirnhaus transformation appears: The Tschirnhaus transformation of the equation for the Tetranacci constant contains only rational coefficients: In this way following expression can be made about the Tetranacci constant: That calculation example however does contain the element of the square root in the Tschirnhaus transformation: In the following we solve a special equation pattern that is easily solvable by using elliptic functions: These are important additional informations about the elliptic nome and the mentioned Jacobi theta function: Computation rule for the mentioned theta quotient: Accurately the Jacobi theta function is used for solving that equation.
In their essay they constructed the quartic Tschirnhaus key in this way: In order to do the transformation Adamchik and Jeffrey constructed equation system that generates the coefficients of the cubic, quadratic and absolute term of the Tschirnhaus key: And for receiving the coefficient of the linear term this cubic equation shall be solved successively: The solution of that system then has to be entered in that mold here: The coefficients Lambda and My can be found out by doing a polynomial division of z^5 divided by the initial principal polynome and reading the resulting remainder rest.
So a Bring Jerrard equation appears that contains only the quintic, the linear and the absolute term.
For doing this, following elliptic modulus or numeric eccentricity and their Pythagorean counterparts and corresponding elliptic nome should be used in relation to Lambda and My after the essay Sulla risoluzione delle equazioni del quinto grado from Charles Hermite and Francesco Brioschi and the recipe on page 258 accurately: These are the elliptic moduli and thus the numeric eccentricities: With the abbreviations ctlh abd tlh the Hyperbolic Lemniscatic functions are represented.
Along with the Abel Ruffini theorem the following equations are examples that can not be solved by elementary expressions, but can be reduced[10] to the Bring Jerrard form by only using cubic radical elements.
To do this on the given principal quintics, we solve the equations for the coefficients of the cubic, quadratic and absolute term of the quartic Tschirnhaus key after the shown pattern.
By doing a polynomial division on the fifth power of the quartic Tschirnhaus transformation key and analyzing the remainder rest the coefficients of the mold can be determined too.