##
**Lectures on forms in many variables.**
*(English)*
Zbl 0185.08304

Mathematics Lecture Note Series. New York-Amsterdam: W.A. Benjamin, Inc. ix, 167 p. (1969).

There are many contributions by many mathematicians on the existence of non-trivial solutions of a homogeneous equation when the number of variables is large enough compared to its degree, and one typical contribution is the theory of \(C_i\)-fields formulated by Lang. The article gives a good treatment of such existence problem.

The text begins with a historical survey (Chapter 1), then gives the Chevalley-Warning theorem which asserts that a finite field is \(C_1\) (Chapter 2). Then the cases of function fields and complete valuation rings are discussed and there are results heavily due to Lang (Chapter 3–6). Then the case of a \(p\)-adic number field is discussed in Chapter 7; there is explained a counter-example to a conjecture of Artin that a \(p\)-adic number field is \(C_2\) (by Terjanian and Schanuel). In Chapter 8, the author exposes a theorem of Brauer and Birch, which asserts a similar result as in the \(C_i\)-field case but without assuming \(C_i\)-property and good for forms of odd degrees. Finally in Chapter 9, recent works of Ax, Kochen and Paul Cohen are discussed.

The text begins with a historical survey (Chapter 1), then gives the Chevalley-Warning theorem which asserts that a finite field is \(C_1\) (Chapter 2). Then the cases of function fields and complete valuation rings are discussed and there are results heavily due to Lang (Chapter 3–6). Then the case of a \(p\)-adic number field is discussed in Chapter 7; there is explained a counter-example to a conjecture of Artin that a \(p\)-adic number field is \(C_2\) (by Terjanian and Schanuel). In Chapter 8, the author exposes a theorem of Brauer and Birch, which asserts a similar result as in the \(C_i\)-field case but without assuming \(C_i\)-property and good for forms of odd degrees. Finally in Chapter 9, recent works of Ax, Kochen and Paul Cohen are discussed.

Reviewer: Masayoshi Nagata (Kyoto)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11D72 | Diophantine equations in many variables |

11D88 | \(p\)-adic and power series fields |

11E99 | Forms and linear algebraic groups |