UDC 513.83 + 519.48

MATHEMATICS

A. F. MUTYLIN

ON THE EMBEDDING OF DISCRETE FIELDS IN CONNECTED ONES

(Presented by Academician A. I. Mal’tsev on 5 X 1965)

We begin with an auxiliary construction. Denote by (C) the interval of the real axis ([0,1]); by (\Omega), the set of all equivalence relations on (C) such that every class is finite and all classes, except for a finite number, are singleton. We shall denote elements of (\Omega) by lowercase letters of the Greek alphabet. We shall say that (\alpha \geqslant \beta) if (a\beta b) implies (a\alpha b). As is easy to verify, (\Omega), with respect to this ordering, is a lattice.

By the length of an (\alpha)-class (M) we shall mean (\max M-\min M), and denote it by (\operatorname{long} M). By the length of (\alpha) we shall mean the sum of the lengths of the (\alpha)-classes (all of them, except for a finite number, are equal to (0)).

Lemma 1. Let (M) be an (\alpha\cup\beta)-class; (P_1,P_2,\ldots,P_m) and (Q_1,Q_2,\ldots,Q_n) all the nonsingleton (\alpha)- and (\beta)-classes contained in it, respectively. Then

[
\operatorname{long} M \leqslant \sum_{i=1}^{m}\operatorname{long} P_i+\sum_{i=1}^{n}\operatorname{long} Q_i.
]

Proof. Suppose the contrary. Denote ([\min M;\max M]=\mathrm{M}), ([\min P_i;\max P_i]=\Pi_i), and ([\min Q_i;\max Q_i]=\Psi_i), where the lengths of (\mathrm{M}), (\Pi_i), and (\Psi_i) are equal to the lengths of the corresponding classes. From our supposition it follows that there exists (x\in \mathrm{M}) such that

[
x\notin \bigcup_{i=1}^{m}\Pi_i\cup\bigcup_{i=1}^{n}\Psi_i.
]

Define (a\gamma b) in each of the following three cases:

a) (a(\alpha\cup\beta)b,\ a\notin M,\ b\notin M);

b) (a,b\in M,\ a<x,\ b<x);

c) (a,b\in M,\ a>x,\ b>x).

It is not hard to verify that (\gamma\in\Omega), (\gamma\geqslant\alpha), (\gamma\geqslant\beta), (\gamma\leqslant\alpha\cup\beta), whence (\gamma=\alpha\cup\beta). But (\min M\,\gamma\,\max M) does not hold, whereas (\min M(\alpha\cup\beta)\max M) does hold, and the lemma is proved.

Since every (\alpha)-class and every (\beta)-class is contained in only one (\alpha\cup\beta)-class, we have

[
\operatorname{long}(\alpha\cup\beta)\leqslant \operatorname{long}\alpha+\operatorname{long}\beta .
\tag{1}
]

Denote by (\alpha^*) the following transformation of (C) into itself:

[
x\alpha^*=\min_{x\alpha y} y.
]

Lemma 2. If (\alpha\leqslant\beta), then (\alpha^\beta^=\beta^*).

Proof. From (x\alpha^*\alpha x) we obtain

[
x\alpha^\beta^=\min_{x\alpha^\beta y}y=\min_{x\beta y}y=x\beta^.
]

Let (P) be an arbitrary field. Denote by (P') the ring of polynomials over (P) in the indeterminates (s_x), where (x) ranges over (C). Denote by (P^0) the field of rational functions over (P) in the same indeterminates. Denote by (\alpha') the transformation of the ring (P') consisting in replacing, in the notation of a polynomial, the indeterminates (s_x) by (s_{x\alpha^*}). It is easy to verify that (\alpha') is a homomorphism.

By (\alpha^0) we shall denote the transformation, defined on those (f\in P^0) for which the denominator of (f) in the canonical representation is not mapped by the transformation (\alpha') to (0), and consisting in the following: if (f) is written in the form (\sum_{i=1}^n \frac{a_i}{b_i}), where (a_i,b_i\in P') and (b_i\alpha'\ne0), then set

[
f\alpha^0=\sum_{i=1}^n \frac{a_i\alpha'}{b_i\alpha'}.
]

Let us prove the correctness of the last definition. Suppose

[
\sum_{i=1}^m \frac{a_i}{b_i}=\sum_{i=1}^n \frac{c_i}{d_i}.
]

Then

[
\sum_{i=1}^m\left(a_i\prod_{\substack{j=1\ j\ne i}}^m b_j\prod_{k=1}^n d_k\right)
=
\sum_{i=1}^n\left(c_i\prod_{\substack{k=1\ k\ne i}}^n d_k\prod_{j=1}^m b_j\right),
]

whence we have the same equality with (\alpha') appended after each polynomial, from which, after division by the product of all (b_j\alpha') and (d_k\alpha'), we obtain the equality

[
\sum_{i=1}^m \frac{a_i\alpha'}{b_i\alpha'}
=
\sum_{i=1}^n \frac{c_i\alpha'}{d_i\alpha'}.
]

Let (p\in P). Then

[
p\alpha^0=\left(\frac{p}{1}\right)\alpha^0=\frac{p\alpha'}{1\alpha'}=\frac{p}{1}=p.
]

The length of a polynomial (f) ((\operatorname{long} f)) will mean

[
\min_{\substack{f\alpha'=0}}[\inf \operatorname{long}\alpha;\,1]^*.
]

The length of a polynomial not equal to (0) is, obviously, not less than the minimal distance between the lower indices of the elements (s_x) occurring in the expression of this polynomial, i.e., from (\operatorname{long} f=0) it follows that (f=0). We note one more obvious property of length:

[
\operatorname{long}(fg)=\min[\operatorname{long} f;\operatorname{long} g].
\tag{2}
]

The weight of the expression (\sum_{i=1}^n \frac{a_i}{b_i}), where (a_i,b_i\in P'), (b_i\ne0), will mean

[
\sum_{i=1}^n \frac{\operatorname{long} a_i}{\operatorname{long} b_i}.
]

The norm of a rational function (f) will mean the exact lower bound of the weights of its expressions, and we shall denote it by (|f|). Let us study the properties of this norm. The first two properties need no proof:

(1^\circ.\ |-f|=|f|.)

(2^\circ.\ |f+g|\le |f|+|g|.)

Let

[
f=\sum_{i\in I}\frac{a_i}{b_i}
\quad\text{and}\quad
g=\sum_{j\in J}\frac{a_j}{b_j}.
]

Denote the weight of the first expression by (p), and that of the second by (q). Order (I\cup J) arbitrarily, but so that from (k<l) (where (k,l\in I\cup J)) it would follow that (\operatorname{long} b_k\le \operatorname{long} b_l). We have

[
fg=
\sum_{k\in I}\left(\frac{a_k}{b_k}\sum_{\substack{l\in J\ l>k}}\frac{a_l}{b_l}\right)
+
\sum_{k\in J}\left(\frac{a_k}{b_k}\sum_{\substack{l\in I\ l>k}}\frac{a_l}{b_l}\right).
]

After bringing the expression in parentheses to a common denominator, the length of its denominator will, by (2), be equal to (\operatorname{long} b_k), while the length of the numerator, by the same equality, will be no greater than (\operatorname{long} a_k); i.e., the weight of such an expression is less than or equal to

[
\sum_{k\in I\cup J}\frac{\operatorname{long} a_k}{\operatorname{long} b_k}=p+q,
]

whence

[
3^\circ.\ |fg|\le |f|+|g|.
]

Let (f) and (g) be arbitrary elements of (P^0), let (\frac{a}{b}) be an irreducible expression of the first of them and (\sum_{i\in I}\frac{a_i}{b_i}), where (a_i,b_i\in P'), an arbitrary expression of the second—

(*) Here the infimum of the empty set is understood to be (+\infty).

Therefore. By (2) we have
[
\frac{\operatorname{long}(aa_i)}{\operatorname{long}(bb_i)}
\leq
\frac{\operatorname{long} a_i}{\operatorname{long} b\,\operatorname{long} b_i},
]
whence
[
\sum_{i\in I}\frac{\operatorname{long}(aa_i)}{\operatorname{long}(bb_i)}
\leq
\frac{1}{\operatorname{long} b}\sum_{i\in I}\frac{\operatorname{long} a_i}{\operatorname{long} b_i}.
]

In view of the arbitrariness of the expression for (g), we have
[
|fg|\leq \frac{1}{\operatorname{long} b}|g|,
]
i.e.

(4^\circ.\ |fg|\leq C_f|g|), where (C_f) depends only on (f).

Let (|f|<1). Then there exists a representation
[
f=\sum_{i=1}^{n}\frac{a_i}{b_i},
]
where (a_i, b_i\in P'), such that
[
\sum_{i=1}^{n}\frac{\operatorname{long} a_i}{\operatorname{long} b_i}=p<1.
]
Hence (\operatorname{long} a_i<1) for all (i). Thus there exist (\alpha_i) such that (a_i\alpha_i'=0) and
[
\operatorname{long}\alpha_i<\frac{\operatorname{long} a_i}{p}.
]
We may assume that, for (i<j), one has (\operatorname{long} b_i\leq \operatorname{long} b_j); otherwise we interchange the summands. Put
[
\bigcup_{i=1}^{k}\alpha_i=\beta_k;\qquad
\beta_1^0\beta_2^0\ldots\beta_k^0=\Gamma_k.
]
Since from (\xi^\eta^=\zeta^) it follows that (\xi'\eta'=\zeta'), by Lemma 2 we obtain
[
\beta_1'\beta_2'\ldots\beta_k'=\beta_k' .
]
By induction it is easily proved that
[
\Gamma_k=\sum_{i=k+1}^{n}\frac{a_i\beta_k'}{b_i\beta_k'}
]
(the transformation (\beta_{k+1}') is applicable to (f\Gamma_k), because, by (1), we have
[
\operatorname{long}\beta_{k+1}\leq\sum_{i=1}^{k}\operatorname{long}\alpha_i
<
\frac{1}{p}\sum_{i=1}^{k}\operatorname{long}a_i
\leq
\frac{1}{p}\operatorname{long}b_k\sum_{i=1}^{k}\frac{\operatorname{long}a_i}{\operatorname{long}b_i}
\leq
\operatorname{long}b_k\leq\operatorname{long}b_l,
]
where (l>k)). Hence (f\Gamma_n^*=0). If (r\in P), then (r\Gamma_n=r), i.e. from (|f|<1) it follows that (f\in P\setminus{0}).

Denote by (K) the set of those (f\in P^0) for which (|f|=0). By (1^\circ, 2^\circ) and (4^\circ), we have that (K) is an ideal. But (P^0) is a field, (|0|=0) and (|1|\geq1). Hence (K={0}). We have obtained the last property of the norm:

(5^\circ.\ |f|=0) if and only if (f=0).

Definition. A topological ring is called a semitopological field if it is a field.

Denote by (U_a), where (a) is a positive real number, the set of all (f) such that (|f|<a). Then from (1^\circ)—(5^\circ) it follows that the (U_a) form a fundamental system of neighborhoods of (0) of a certain semitopologization of the field (P^0), i.e. (P^0) is a semitopological field. Denote by (\xi_{ab}), where (a,b\in C), the equivalence relation with the unique non-one-element class ({a,b}). Then
[
\operatorname{long}\xi_{ab}=|a-b|
]
and
[
(s_a-s_b)\xi'{ab}=s=0,}-s_{\min[a,b]
]
i.e.
[
|s_a-s_b|\leq |a-b|;
]
in other words, the topology on (s_C) is weaker than or the same as the one induced from (C) under the mapping (x\mapsto s_x). But the topology of (C) is connected. Hence (s_C) is all the more connected; and if in a semitopological field there exists a connected non-one-element set, then the field itself is connected, since by a nondegenerate linear transformation any pair of distinct points is carried into any pair of distinct points, while a nondegenerate linear transformation in a semitopological field is a homeomorphism.

We shall give a new proof of Theorem 9 from (1):

Lemma 3. The topology of a semitopological field is weakened to the topology of a topological field.

Proof. Choose any fundamental system of symmetric (i.e. containing (-x) together with (x)) neighborhoods of (0) of the semitopological

* But (\Gamma_k\ne \beta_k^0), in general, since from (\xi^\eta^=\zeta^*) it does not follow that (\xi^0\eta^0=\zeta^0) (the domain of definition may change).

of the topological field ({U_\alpha}). Consider the system of symmetric sets

[
\frac{U_\alpha}{1+(U_\alpha\setminus{-1})}.
]

Having verified for it the required properties, we are convinced that this is a fundamental system of neighborhoods of (0) for some field topologization. As an example, let us verify separability. Let (x\ne 0); then there exists (U_\alpha\not\ni x). There exists (U_\beta) such that (U_\beta+U_\beta\subseteq U_\alpha). There exists (U_\gamma) such that (xU_\gamma\subseteq U_\beta) and (U_\gamma\subseteq U_\beta). Then (U_\gamma+xU_\gamma\not\ni x), whence, by symmetry,

[
(x+xU_\gamma)\cap U_\gamma=\varnothing,
]

or

[
{x}\cap\frac{U_\gamma}{1+(U_\gamma\setminus{-1})}=\varnothing,
]

as was required to prove. Since

[
\frac{U_\alpha}{1+(U_\alpha\setminus{-1})}\supset U_\alpha,
]

the new topology is the same or weaker.

Apply the lemma just proved to (P^0), to the system ({U_\alpha}). We obtain some field topologization (\tau). It is connected, since it is weaker than, or the same as, the original one. In conclusion, let us show that (P) is discrete in (P^0_\tau). From (2) it follows that, for (a\in P\setminus{0}) and (f\in P'), we have

[
\operatorname{long}(af)=\min[\operatorname{long}a;\operatorname{long}f]\le \operatorname{long}f
=\operatorname{long}(a^{-1}af)=\min[\operatorname{long}a^{-1};\operatorname{long}(af)]\le \operatorname{long}(af),
]

i.e. (\operatorname{long}f=\operatorname{long}(af)), whence it follows that (|ax|=|x|), if (a\in P\setminus{0}). Hence

[
(P\setminus{0})U_{1/2}\subseteq U_{1/2}.
]

By the property obtained in the proof of (5^\circ),

[
(P\setminus{0})\cap U_1=\varnothing,
]

and hence

[
\frac{U'{1/2}}{1+U\cap(P\setminus{0})=\varnothing,}
]

and, therefore, (P) is discrete. Thus, the following has been proved.

Theorem. Every discrete field can be embedded in a connected one.

Corollary. There exists a connected field of any characteristic.

The latter solves the problem posed by I. R. Shafarevich. I take this opportunity to express my gratitude to Prof. L. A. Skornyakov for valuable comments on the manuscript.

Moscow State University
named after M. V. Lomonosov

Received
27 IX 1965

REFERENCES

1. B. Gelbaum, G. K. Kalisch, J. M. H. Olmsted, Proc. Am. Math. Soc. 2, No. 5, 807 (1951).