N – real fields
A field F is n-real if −1 is not the sum of n squares in F. It is shown that a field F is m-real if and only if rank (AAt ) = rank (A) for every n × m matrix A with entries from F. An n-real field F is n-real closed if every proper algebraic extension of F is not n-real. It is shown that if a 3...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2003 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2003
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| Цитувати: | N – real fields / S. Feigelstock // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 1–6. — Бібліогр.: 8 назв. — англ. |
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Feigelstock, S. 2019-06-17T10:42:57Z 2019-06-17T10:42:57Z 2003 N – real fields / S. Feigelstock // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 1–6. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 12D15. https://nasplib.isofts.kiev.ua/handle/123456789/155693 A field F is n-real if −1 is not the sum of n squares in F. It is shown that a field F is m-real if and only if rank (AAt ) = rank (A) for every n × m matrix A with entries from F. An n-real field F is n-real closed if every proper algebraic extension of F is not n-real. It is shown that if a 3-real field F is 2-real closed, then F is a real closed field. For F a quadratic extension of the field of rational numbers, the greatest integer n such that F is n-real is determined. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics N – real fields Article published earlier |
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N – real fields |
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N – real fields Feigelstock, S. |
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N – real fields |
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N – real fields |
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N – real fields |
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N – real fields |
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n – real fields |
| author |
Feigelstock, S. |
| author_facet |
Feigelstock, S. |
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2003 |
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English |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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Article |
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A field F is n-real if −1 is not the sum of n
squares in F. It is shown that a field F is m-real if and only
if rank (AAt
) = rank (A) for every n × m matrix A with entries
from F. An n-real field F is n-real closed if every proper algebraic
extension of F is not n-real. It is shown that if a 3-real field F
is 2-real closed, then F is a real closed field. For F a quadratic
extension of the field of rational numbers, the greatest integer n
such that F is n-real is determined.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155693 |
| citation_txt |
N – real fields / S. Feigelstock // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 1–6. — Бібліогр.: 8 назв. — англ. |
| work_keys_str_mv |
AT feigelstocks nrealfields |
| first_indexed |
2025-11-24T06:35:03Z |
| last_indexed |
2025-11-24T06:35:03Z |
| _version_ |
1850843108221124608 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2003). pp. 1 – 6
c© Journal “Algebra and Discrete Mathematics”
N – real fields
Shalom Feigelstock
Communicated by M. Ya. Komarnytskyj
Abstract. A field F is n-real if −1 is not the sum of n
squares in F . It is shown that a field F is m-real if and only
if rank (AAt) = rank (A) for every n × m matrix A with entries
from F . An n-real field F is n-real closed if every proper algebraic
extension of F is not n-real. It is shown that if a 3-real field F
is 2-real closed, then F is a real closed field. For F a quadratic
extension of the field of rational numbers, the greatest integer n
such that F is n-real is determined.
A field F is formally real if −1 is not a sum of squares in F , or
equivalently if 0 is not a sum of squares in F with non-zero summand. The
study of these fields was initiated by Artin and Schreier, [1]. Many results
on vector spaces over a subfield of the field of real numbers remain valid
if the field of scalars is formally real; e.g. many results on real quadratic
forms. For finite dimensional vector spaces the following weaker condition
often suffices:
Definition. Let n be a positive integer, and let ν = (a1, . . . , an), ω =
(b1, . . . , bn) ∈ Fn. The scalar product ν ·ω = a1b1 + . . .+ anbn. A field F
is n-real if for every non zero-vector ν ∈ Fn the scalar product ν · ν 6= 0.
Clearly every field is 1-real. For n > 1, a field F is n-real if and only
if −1 is not the sum of n− 1 squares, and F is a formally real field if and
only if F is n-real for every positive integer n. If F is n-real then F is
m-real for every m < n.
Example. The field Q(
√
−5) is 2-real but not 3-real.
2000 Mathematics Subject Classification: 12D15.
Key words and phrases: n-real, n-real closed.
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.2 N – real fields
Theorem 1. A field F is m-real if and only if for every positive integer n,
and every n×m matrix A with entries from F , the rank r(AAt) = r(A).
Proof. Suppose that F is m-real, and let r(A) = k. Performing a Gram-
Schmidt like process on k linearly independent rows of A, with the scalar
product replacing the inner product, yields orthogonal vectors ν1, ..., νk ∈
Fm . The n×m matrix B with first k rows ν1, ..., νk , and with remain-
ing n − k rows the 0-vector, can be obtained by performing a series of
elementary row operations on A. Therefore there exists an n × n matrix
C with entries from F such that B = CA. The matrix BBt is diagonal
with first k diagonal entries non-zero, and remaining entries 0. Therefore
k = r(BBt) ≤ r(AAt) ≤ r(A) = k, and so r(AAt) = r(A). If F is not
m-real then there exists a non-zero vector ν ∈ Fm such that ν · ν = 0.
For any positive integer n let A be the n × m matrix all of whose rows
are ν. Then r(A) = 1, but r(AAt) = 0.
If every proper algebraic extension of a formally real field F is not
formally real, then F is said to be a real closed field. The obvious parallel
concept for n-real fields is:
Definition. An n-real field F is an n-real closed field if every proper
algebraic extension of F is not n-real.
A simple Zorn’s Lemma argument yields:
Lemma 2. Every n-real field, n > 1, is contained in an n-real closed
field.
Lemma 3. Let F be a 2-real closed field and let a ∈ F . Then either a
or −a is a square in F .
Proof. If a is not a square in F then F (
√
a) is not 2-real, so there exist
b, c ∈ F such that (b + c
√
a)2 = −1, i.e., b2 + c2a + 2bc
√
a = −1. Since√
a /∈ F , and F is 2-real it follows that b = 0, and −a = (c−1)2.
Recall [4], p. 271, that a field F is ordered if there exists a subset P
of F such that F = P
⋃
{0}
⋃
−P is a disjoint union, and a + b, ab ∈ P
for all a, b ∈ P . A well known result of Artin-Schreier is that a field is
formally real if and only if it is ordered.
Corollary 4. Let F be a 2-real closed field. If F is 3-real then F is real
closed.
Proof. Let F be a 2-real closed, 3-real field. It suffices to show that F
is ordered. Let P be the set of non-zero squares in F . It follows from
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.S. Feigelstock 3
Lemma 3 that F = P
⋃{0}⋃−P . If a ∈ P
⋂−P , then there exist non-
zero elements b, c ∈ F such that a = b2 = −c2, and so b2 + c2 = 0, a
contradiction. Therefore the above union is a disjoint union. Let a, b ∈ P .
Clearly ab ∈ P , so it suffices to show that a + b ∈ P . If not, then by
Lemma 3 there exists c ∈ F such that a + b = −c2. Since a, b ∈ P there
exist a1, b1 ∈ F such that a = a2
1, and b = b2
1. Therefore a2
1 + b2
1 + c2 = 0
contradicting the fact that F is 3-real.
If F is a real closed field, and f(x) ∈ F [x] is a polynomial of odd
degree, then f(x) has a root in F ; see [8], p. 226, Theorem 2. An almost
identical argument yields:
Theorem 5. If F is an n-real closed field, n > 1, and f(x) ∈ F [x] is a
polynomial of odd degree, then f(x) has a root in F .
Let F be a field of prime characteristic p. Since 0 is the sum of p
copies of 12 it follows that F is not formally real. The following known
number theory result yields that properties of p determine completely
whether or not F is n-real for every positive integer n.
Proposition 6. Let n be a positive integer.
1) n is the sum of two squares of integers if and only if the prime
decomposition of n has no factor of the form qe, with q a prime satisfying
q ≡ 3 mod 4, and e odd.
2) n is not the sum of three squares of integers if and only if n =
4m(8k + 7), with m, k non-negative integers.
Proof. See [5], p. 110 Corollary 5.14, and [7], p. 45, Theorem (Gauss).
Theorem 7. A field F of prime characteristic p is not 3-real. It is 2-real
if and only if p ≡ 3 mod 4.
Proof. Since 12 + 12 ≡ 0 mod 2 it may be assumed that p is odd. If
p.7 mod 8 then p is the sum of three squares of integers by Proposition
6.2, so F is not 3 - real. If p ≡ 7 mod 8 then 2p ≡ 6 mod 8 so 2p is the
sum of three squares of integers by Proposition 6.2 which yields that F is
not 3-real. The field F is 2-real if and only if −1 is a quadratic nonresidue
mod p, which occurs if and only if p ≡ 3 mod 4.
A well known result of Lagrange is that every positive integer is the
sum of 4 squares of integers. This yields:
Lemma 8. Let F = Q(
√
a), α ∈ Q be a quadratic extension of the field
of rational numbers. If F is not real then F is not 5-real.
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.4 N – real fields
Proof. If F is not real then it may be assumed that α is a negative integer,
[5], Theorem 9.20. Since −α is the sum of 4 squares of integers, it follows
that F is not 5-real.
Definition. Let F be a field which is not formally real. The least positive
integer n such that −1 is the sum of n squares in F was called the Stuffe
of F by Pfister, [6]; it is, of course, the greatest positive integer n such
that F is n-real.
Pfister proved the following:
Proposition 9. Let n be a positive integer. There exists a field with
Stuffe n if and only if n = 2k, with k a non-negative integer.
Proof. See [6], Satz 4 and Satz 5.
Lemma 8 and Proposition 9 yield:
Corollary 10. Let F be a quadratic extension of Q. If F is not real then
the Stuffe of F is either 1, 2, or 4.
Fein, Gordon and Smith proved the following:
Proposition 11. For m a negative square free integer, −1 is the sum of
two squares in Q(
√
m) if and only if m ≡ 2 or 3 mod 4, or m ≡ 5 mod 8.
Proof. [3] Theorem 7.
Since every imaginary quadratic extension of Q is of the form Q(
√
m),
with m a square free negative integer, Corollary 10 and Proposition 11
completely determine the Stuffe of such extensions as follows:
Theorem 12. For m a square free negative integer the Stuffe of Q(
√
m)
is :
1 if m = −1,
2 if m ≡ 2 or 3 mod 4, or if m ≡ 5 mod 8, and
4 otherwise.
Example. The Stuffe of Q(
√
−7) is 4.
If A is a commutative ring and if a, b ∈ A are both the sum of four
squares in A, then an equality of Euler, [5], Lemma 5.3, yields that ab
is the sum of four squares in A. The following generalization of Euler’s
result for fields was proved by Pfister.
Proposition 13. Let F be a field, and let n = 2m, with m a non-negative
integer. If a, b ∈ F are both the sum of n squares in F then ab is the sum
of n squares in F .
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.S. Feigelstock 5
Proof. See [6], Satz 2.
Corollary 14. Let F be a field extension of Q. If F is formally real then
a ∈ F is the sum of four squares in F if and only if a ≥ 0. If F is not
formally real, then the Stuffe of F is ≤ 4 if and only if every rational
number is the sum of four squares in F .
Proof. Every positive integer is the sum of four squares of integers, [5],
Theorem 5.6. If a non-zero element a in a field E is the sum of n squares
in E, then it is readily seen that a−1 is the sum of n squares in E.
Therefore either Euler’s equality, or Proposition 13 yield that every non-
negative rational number is the sum of four squares of rational numbers.
If a negative rational number a is the sum of four squares in F then
−1 = a(1/ |a|) is the sum of four squares and the Stuffe of F is ≤ 4.
Conversely, if the Stuffe of F is ≤ 4 then every rational number is the
sum of four squares in F by Proposition 13.
The following Proposition combines two results of Cassels:
Proposition 15. Let F be a field with characteristic 6= 2, let a ∈ F , and
let x be an indeterminant. Then x2 + a is the sum of n > 1 squares in
F [x] if and only if either −1 or a is the sum of n − 1 squares in F .
Proof. See [2], Theorem 2.
A simple consequence of Proposition 15 is:
Corollary 16. If F is a non-formally real field with characteristic 6= 2,
and with Stuffe n, then every element in F is the sum of n + 1 squares
in F .
Proof. Let a ∈ F . By Proposition 15, there exist
pi(x) ∈ F [x], i = 1, ..., n + 1,
such that x2 + a =
∑
n+1
i=1
pi(x)2, so a =
∑
n+1
i=1
pi(0)2.
References
[1] E. Artin and O. Schreier, Algebraische Konstructionen reeler Korper, Abh. Math.
Sem. Hamburg, 5 (1926), 83-115.
[2] J. W. S. Cassels, On the representation of rational functions as sums of squares,
Acta Arith. 9 (1964), 79-82.
[3] B. Fein, B. Gordon, and J. H. Smith, On the representation of -1 as a sum of two
squares in an algebraic number field, Journal of Number Theory 3(1971), 310-315.
[4] S. Lang, Algebra, 1
st edition, Addison-Wesley, Reading, 1965.
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.6 N – real fields
[5] I. Niven and H. Zuckerman, An Introduction to the Theory of Numbers, Wiley,
New York, 1960.
[6] A. Pfister, Darstellung von -1 als Summe von Quadraten in einen Korper, J. Lon-
don Math. Soc., series 1, 40 (1965), 159-165.
[7] J. P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer,
New York, 1973.
[8] B. L. Van der Waerden, Modern Algebra, Vol. 1, Ungar, New York, 1964.
Contact information
S. Feigelstock Department of Mathematics Bar-Ilan, Uni-
versity Ramat Gan, Israel
E-Mail: feigel@macs.biu.ac.il
Received by the editors: 03.03.2003
and final form in 23.10.2003.
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