Geodesic
On the extension of the ring topology of a~$\sigma$-bounded field to a~simple transcendental extension of the field
Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 271-287

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If $\tau$ is a ring topology of a field $R$ such that $(R,\tau)$ is the union of countably many bounded sets, then there exists a ring topology $\hat\tau$ on a simple transcendental extension $R[x]$ of $R$ such that $(R[x],\hat\tau)$ is the union of countably many bounded sets and $\tau$ is the restriction of $\hat\tau$ to $R$. Bibliography: 6 titles. 
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     author = {V. I. Arnautov},
     title = {On the extension of the ring topology of a~$\sigma$-bounded field to a~simple transcendental extension of the field},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_61_2_a0/}
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V. I. Arnautov. On the extension of the ring topology of a~$\sigma$-bounded field to a~simple transcendental extension of the field. Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 271-287. http://geodesic.mathdoc.fr/item/SM_1988_61_2_a0/