On the solvability of equations of higher degree (Q2759835)

The author shows that the solvability of the general polynomial, the formula to solve all polynomials, and the solvability of all polynomials are distinct, though very near to each other.NEWLINENEWLINENEWLINELet \(K\) be a field. The polynomials of a given degree \(n\) in \(K[x]\) are said to be solvable if each of the polynomials having degree \(\leq n\) is solvable. \(K\) is said to have property \(A_n\) if all polynomials in \(K[x]\) of degree \(\leq n\) are solvable.NEWLINENEWLINENEWLINELet \(\mathbb{Q}_p\) be the prime subfield of \(K\), where either \(p=0\) or \(p\) is a prime. The general polynomial over \(K\) is \(F_y(x) = x^n +y_{n-1} x^{n-1} +\cdots + y_1x +y_0\), where \(y_{n-1},\dots,y_1,y_0\) are algebraically independent indeterminates over \(\mathbb{Q}_p\). The general polynomial \(F_y(x)\) of degree \(n\) is called solvable if its zeros are contained in a radical extension. \(K\) is said to have property \(C_n\) if the general polynomial \(F_y(x)\) of degree \(n\) is solvable.NEWLINENEWLINENEWLINEThere is said to be a solving formula for polynomials of degree \(\leq n\) over \(K\) if \(\mathbb{Q}_p(y_{n-1},\dots,y_1,y_0)\) has a radical extension \(\mathbb{Q}^2_p\) containing fixed elements \(x_{n-1},\dots,x_1,x_0\) such that the homomorphism \(\varphi : \mathbb{Q}_p[y_{n-1},\dots,y_1,y_0]\to K\) can be extended to an integrally closed subring \(\mathbb{Z}^2_p\), sending \(x_{n-1},\dots,x_1,x_0\) to the zeros of the polynomial \(x^n + a_{n-1} x^{n-1} +\cdots + a_1x + a_0\). Property \(B_n\) is there exists a solving formula for polynomials of degree \(\leq n\) over \(K\).NEWLINENEWLINENEWLINEThe author shows that for fields of characteristic 0, \(A_n\) implies \(C_n\), and for infinite fields, \(B_n\) implies \(C_n\). The author showsNEWLINENEWLINENEWLINE(1) there exist infinitely many fields where \(A_n\) holds but \(B_n\) does not;NEWLINENEWLINENEWLINE(2) there exist infinitely many fields where \(B_n\) holds but \(C_n\) does not;NEWLINENEWLINENEWLINE(3) for any \(k\) there exist \(k\) fields of distinct characteristic where \(B_n\) holds with the same formula, but it is not true for infinitely many fields.NEWLINENEWLINENEWLINEThe author also shows that there exist infinitely many \(n\) such that \(B_n\) does not imply \(B_{n+1}\), and infinitely many \(n\) such that \(A_n\) does not imply \(A_{n+1}\).NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].