Purely inseparable extensions of

Author:
D. Daigle

Journal:
Proc. Amer. Math. Soc. **121** (1994), 1-12

MSC:
Primary 13F20; Secondary 13B02

MathSciNet review:
1227516

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let **k** be a field of characteristic and *R* a polynomial ring in two variables over **k**. Define *weak variable* of *R* to mean an element *u* of *R* such that is irreducible for each and such that for some and some integer . Given a weak variable *u* of *R*, consider all such that for some *n*; if one of these *v* is "absolutely smaller" than *u* (roughly, for *all* coordinate systems (*X*, *Y*) of *R*), we call it an *R*-companion of *u*. The main result gives a connection between the structure of a purely inseparable extension , where *A* is a polynomial ring in two variables, and whether or not there exists a companion for each *u* in a suitable set of weak variables of *R*.

**[1]**S. S. Abhyankar,*Lectures on expansion techniques in algebraic geometry*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, 1977. Notes by Balwant Singh. MR**542446****[2]**Shreeram S. Abhyankar and Balwant Singh,*Embeddings of certain curves in the affine plane*, Amer. J. Math.**100**(1978), no. 1, 99–175. MR**0498566****[3]**D. Daigle,*Plane Frobenius sandwiches of degree 𝑝*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 885–889. MR**1118085**, 10.1090/S0002-9939-1993-1118085-2**[4]**Richard Ganong,*On plane curves with one place at infinity*, J. Reine Angew. Math.**307/308**(1979), 173–193. MR**534219**, 10.1515/crll.1979.307-308.173**[5]**R. Ganong,*Plane Frobenius sandwiches*, Proc. Amer. Math. Soc.**84**(1982), no. 4, 474–478. MR**643732**, 10.1090/S0002-9939-1982-0643732-1**[6]**Nathan Jacobson,*Lectures in abstract algebra*, Springer-Verlag, New York-Berlin, 1975. Volume II: Linear algebra; Reprint of the 1953 edition [Van Nostrand, Toronto, Ont.]; Graduate Texts in Mathematics, No. 31. MR**0369381****[7]**Tzuong Tsieng Moh,*On the classification problem of embedded lines in characteristic 𝑝*, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 267–279. MR**977764****[8]**Peter Russell,*Hamburger-Noether expansions and approximate roots of polynomials*, Manuscripta Math.**31**(1980), no. 1-3, 25–95. MR**576491**, 10.1007/BF01303268

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
13F20,
13B02

Retrieve articles in all journals with MSC: 13F20, 13B02

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1227516-X

Article copyright:
© Copyright 1994
American Mathematical Society