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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] Arnautov V. I., “Prodolzhenie lokalno ogranichennoi topologii polya na ego prostoe transtsendentnoe rasshirenie”, Algebra i logika, 20:50 (1981), 511–521 | MR | Zbl

[2] Arnautov V. I., Vizitiu V. N., “Prodolzhenie lokalno ogranichennoi topologii polya na ego algebraicheskoe rasshirenie”, Izv. AN MSSR. Ser. fiz.-tekh. i matem. nauk, 1974, no. 2, 29–43 | MR | Zbl

[3] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[4] Vizitiu V. N., “O prodolzheniyakh $m$-topologii kommutativnykh kolets”, Algebry i moduli. Matem. issl., 48 (1978), 24–39 | MR | Zbl

[5] Hinrichs L. A., “The existence of topologies of field extensions”, Trans. Amer. Math. Soc., 113:3 (1964), 397–405 | DOI | MR | Zbl

[6] Podewski K. P., “Transcedental extensions of field topologies on countable fields”, Proc. Amer. Math. Soc., 39 (1973), 33–38 | DOI | MR | Zbl