On the degree of irrationality in Noether's problem (Q2810670)

The main result reads as follows: Let \(A\) be a finite abelian group, and let \(K\) be a fixed field such that \([K(\zeta_s):K]\) is cyclic for every prime power \(s\) that divides the exponent of \(A\) and is relatively prime to \(\mathrm{char}(K)\), where \(\zeta_s\) denotes a primitive \(s\)th root of unity. Then the so-called degree of irrationality \(\mathrm{Irr}(K,A)\) is less than some explicit bound involving only the arithmetic of certain cyclotomic number rings (for details see Theorem 5). Note that the quantity \(\mathrm{Irr}(K,A)\) is defined as the minimum degree of \(K(A)\) over any purely transcendental extension field of \(K\).NEWLINENEWLINEFirst the author reviews some results of \textit{H. W. Lenstra jun.} [Invent. Math. 25, 299--325 (1974; Zbl 0292.20010)] needed for proving Theorem 1, but presents Lenstra's method in a form suitably adapted for his own purposes. Then he obtains Theorem 1 in the particular case \(\mathbb{Z}/p^n\mathbb{Z}\), \(p\) an odd prime, of \(A\) (Theorem 4), from which he deduces Theorem 1 for a general abelian group \(A\).