Geodesic
On Rationality of Algebraic Function Fields
Canadian mathematical bulletin, Tome 12 (1969) no. 3, pp. 339-341

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Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case. 
Nobusawa, Nobuo. On Rationality of Algebraic Function Fields. Canadian mathematical bulletin, Tome 12 (1969) no. 3, pp. 339-341. doi: 10.4153/CMB-1969-044-0
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     title = {On {Rationality} of {Algebraic} {Function} {Fields}},
     journal = {Canadian mathematical bulletin},
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     year = {1969},
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     number = {3},
     doi = {10.4153/CMB-1969-044-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-044-0/}
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