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This is based on a talk by Zinovy Reichstein from the PIMS Algebra Summer School in Edmonton.

The motivation comes from looking at ways to simplify polynomials. For example, if we start with a quadratic equation $x^2+ax+b$, we can remove the linear term by setting $y=x+\frac{a}{2}$; our equation then becomes $y^2+b'$.

We can do something similar with any degree polynomial. Consider the polynomial $x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_n$. We may make the substitution $y=x+\frac{a_1}{n}$ to remove the $(n-1)$-degree term.

We can also make the coefficients of the linear and constant terms equal with the substitution $z=\frac{b_n}{b_{n-1}}y$ (where the $b$‘s are the coefficients of the polynomial expressed in terms of $y$).

Enough for motivation. Suppose $f(x)=x^n+a_1x^{n-1}+\cdots+a_n$ is a polynomial, and let $K=\mathbb{C}(a_1,\ldots,a_n)$. (So, in particular, $a_1,\ldots,a_n$ form a transcendence basis for $K$ over $\mathbb{C}$.) Let $L=K[x]/(f(x))$. A Tschirnhaus transformation is an element $y\in L$ so that $L=K(y)$.

Applying a Tschirnhaus transformation to a polynomial of degree $n$ gives us another polynomial of degree $n$ with different coefficients. They also allow us to simplify polynomial expressions in various senses. We will use the following two criteria of simplification:

1) A simplification involves making as many coefficients as possible 0.

2) A simplification involves making the transcendence degree of $\mathbb{C}(b_1,\ldots,b_n)$ over $\mathbb{C}$ as small as possible. (For a polynomial of degree $n$, we will write $d(n)$ for this number.)

Suppose $n=5$. Hermite showed that it is possible to make $b_1=b_3=0$ and $b_4=b_5$. Therefore $d(5)\le 2$. Klein showed that $d(5)$ is in fact equal to 2.

Now suppose $n=6$. Joubert showed that again we can make $b_1=b_3=0$ and $b_5=b_6$. Therefore $d(6)\le 3$.

It is unknown whether we can make $b_1=b_3=0$ when $n=7$. However, it is known that we cannot do so if $n$ is of the form $n=3^r+3^s$ for $r>s\ge 0$ or $n=3^r$ for $r\ge 0$. It is also known (Buhler and Reichstein) that $d(n)\ge\left\lfloor\frac{n}{2}\right\rfloor$.