1. Introduction
For algebraic number theory [1], the overall understanding process starts with defining some basic algebraic number theory annotations, such as algebraic numbers and integers [2], number fields, rings of integers, norm and discriminant, fractional ideals, and class groups and lattices. As one of the most essential parts of algebraic number theory, factorising prime numbers into prime ideals of a ring of integers and applying it in the case of simple number fields (such as quadratic fields) is very important intermediate knowledge. Only after mastering the basic definition and such a decomposition relationship can the following Minkowski’s first theorem be introduced to try to compute class numbers and class groups of simple number fields. This paper can be divided into two parts, describing the process of decomposition of prime ideals and the corresponding applications, respectively. To be more specific, it discusses what kind of prime numbers can be written in the form of a sum of two integer squares, which is a very classical part of algebraic number theory.
The most important part of the first part is that if we give a number field, give the corresponding ring of integers and have a minimal polynomial, then we can perform in the given number field for some given prime numbers. break down. The second part gives more inspiration when we consider an algebraic decomposition problem, in addition to analyzing the properties of the number itself, we can also use the properties of groups, rings, and fields on the basis of group theory. Considering the problem, this actually inspires us to think about the problem from many aspects when facing Fermat's last theorem.
2. Factoring primes
2.1. Some basic definitions
Several concepts of algebraic number theory have a pivotal role [3]: numbers, polynomials, and equations. So this paper gives definitions of some of the core terms before discussing the central issues. First of all, for numbers that people have a lot of contact with, the more special numbers in algebraic number theory are algebraic numbers and transcendental numbers [4]. Based on this background, the following are some definitions.
Definition 2.1. A complex number
is called algebraic if there is a non-zero polynomials
with
. If
is not algebraic, it is called transcendental.
Definition 2.2. The number field generated by an algebraic number
is the smallest subfield of
that contains
. We denote this field by
.
A subfield
is an algebraic number field (or a number field) if
for an algebraic number
.
Definition 2.3. A complex number
is an algebraic integer if there is a monic polynomial
with
.
Definition 2.4. Let
be a number field. The ring
consisting of all algebraic integers in
is called the ring of integers of
, where the set
of algebraic integers is a subring of
.
2.2. Prime ideals
Lemma 2.5. Let
be a number field and
a non-zero prime ideal [5]. Then
is a non-zero prime ideal of
, so
, for a prime
.
Proof. If
,
, then
,
, then
is an ideal of
. If we have
such that
, since
is a prime ideal, then
or
. Since
, we can get
or
, thus
is a prime ideal of
. Let
,
. We konw that
, and
. Hence,
. As a non-zero prime ideal of
, it must have the form
for a prime
.
Definition 2.6. Let
be a prime. The prime ideals
appearing in the prime ideal factorization
of
are said to be lying above
[6].
Lemma 2.7. Let
be a prime,
a prime ideal of
. Then
lies above
if and only if
.
Proof. If
lies above
, it appears in the prime ideal factorisation of 
. Thus, 
. Then,
. Since
is a prime ideal
by Lemma 2.5, with
is a prime, then it can get 
, thus
.
If
, then
, thus
, by the properties of ideals of
: "To contain is to divide", thus
. Therefore, if
is the prime ideal factorisation of
, we must have
for some
.
Theorem 2.8. Let
be a number field and assume that
, for some
. Let
be prime and
a monic polynomial, such that
is irreducible over
and divides
where
is the minimal polynomial. Then the ideal
is a prime ideal of
. Moreover,
lies above
and
.
Proof. We consider the residue class field
. Recall that
. A function
is defined by setting
. If
, then
divides
, thus
where
. By Gauss Lemma, it is found that
, such that
. Since
is monic, then
. However,
implies that
. Thus
where
. Since
, then
, thus
and
is well defined. It is clearly a ring homomorphism and subjective. Consider the ideal
of
. By the First Isomorphism Theorem,
is an isomorphism, then 
is a field with
elements. Therefore,
is a maximal ideal of
and a prime ideal with
.
We just need to show that
. First, since
, we have
. Moreover,
, as
. Thus
. Conversely, let
. Then
, so
, for some
. Then the polynomials
and
are congruent modulo
, so all coefficients of
are divisible by
. Hence, there is a polynomial
, such that
. Plugging in
,
is obtained. Thus
. Since
, we have
. By Lemma 2.5,
is a prime ideal of
that contains
, thus
. By Lemma 2.7, it shows that
lies above
.
2.3. Factoring prime
Theorem 2.9. Let
be a number field and assume that
for some
. Let
be prime and
be the factorization of
[7] into irreducibles. That is,
are monic polynomials, such that the
are distinct and irreducible in
, and
. Then the prime ideals of
lying above
are precisely the ideals
, for
. The ideal
of
factorises into prime ideals as
Proof. It is known from Theorem 2.8 that all
are prime ideals with
, where
. From the factorization
, it can get
since all
are monic,
. For arbitrary ideals
of
, Equation (1) is obtained:
(1)
Applying this inductively to the
, Equation (2) is obtained:
(2)
The last equality holds due to
, as then
divides all coefficients of
, so
, for some
. But then
.
Thus,
is obtained and hence
with
. Hence, the
is indeed the only prime ideals of
lying above
. Moreover, Equation (3) is obtained:
(3)
Compare this to
to conclude that
for all
.
3. Primes as sums of two squares
Lemma 3.1. Let
be an odd prime, such that
, with
. Then 
.
Proof. Since
, 
, 
and
, we found that squares are always congruent to
or
modulo
. If
is a sum of two squares, then
,
,
mod
. Then
is impossible and the only prime congruent to
modulo
is
.
The less trivial reverse direction will be proved, showing that every prime
can be written as
, with
. This will be done by factoring the ideal
in the ring of integers
of
. The factorisation depends on the roots of
in
.
Lemma 3.2. Let
be a prime. Then
is a square modulo
, i.e., there is
with
.
Proof. Let
be a generator of
, then
and write
. By Fermat's Little Theorem, then
. The polynomial
has only two roots 
. Thus
. If
, so
can not generate all of
, as then 
, so the powers
, for
capture only
elements of
. Hence,
and thus,
is a square modulo
.
Theorem 3.3. Let
be an odd prime. Then
is a sum of two squares if and only if 
[8,9].
Proof. From Lemma 3.1, it is known that the "only if" part has been proved, so let us prove the "if" part. Let
, then
. Consider the factorisation of the ideal
and note that
. By Lemma 3.2, there is
, such that
. Thus Equation (4) is obtained:
(4)
If
, then
. Since
, so there is only one possible value: 
, but that is impossible because
, thus
, so these are two distinct roots. Hence,
is split in
and
, for two distinct prime ideals
with
. Since every ideal of
is principle. Hence,
, for some 
. Since
and
, then Equation (5) is obtained:
(5)
4. Conclusion
This paper describes the process of decomposition of prime ideals and the corresponding applications: i.e., it discusses what kind of prime numbers can be written in the form of a sum of two integer squares and we got the result that for odd prime
, it is a sum of two squares if and only if 
,this part is a very classical part of algebraic number theory, on the basis of which if Minkowski bound and class groups, etc. are introduced, the property of the unique factorization domain (UFD) can be used to solve some problems about the Diophantine equations [10]. For instance, it can be shown that there are no 
with 
, which in the long run will be useful for the proof of Fermat's Last Theorem.
In the proof of the theorem, we use the fact that every ideal in
is principle, we did not present the proof of this result in the article due to space constraints. Also, Theorem 2.9 can be applied to many number fields, but not all of them, that is because the assumption 
, for some algebraic integer
. But for many number fields, such an
does not exist, we just found such
for quadratic and cyclotomic fields. Therefore, the results given in this article actually have certain limitations. On the basis of this problem, we can gradually try to use the idea of decomposition to consider proof of the special case of the Fermat’s last theorem for regular primes.
References
[1]. Bhaskar, J. (2008). Sum of two squares. https://www.math.uchicago.edu/~may/VIGRE/ VIGRE2008/REUPapers/Bhaskar.pdf.
[2]. Homeworkhelp. Com and Inc. Factoring and Primes (High School Math).
[3]. Stewart, I. and Tall, D. (2001). Algebraic Number Theory and Fermat's Last Theorem: Third Edition. A K Peters/CRC Press. ISBN-10: 1568811195. ISBN-13: 978-1568811192.
[4]. Edwards, H. M. (1977). Fermat's last theorem: A genetic introduction to algebraic number theory. In: Graduate Texts in Mathematics. Springer New York, NY.
[5]. Stein, M. R. and Dennis, R. K. (1989). Algebraic K-Theory and Algebraic Number Theory: Comtemporar Math., 83. American Mathematical Society, Providence.
[6]. Rosen, K. H. (2000). Elementary number theory and its applications. Addison Wesley. ISBN-10: 0201870738. ISBN-13: 978-0201870732.
[7]. Zagier, D. (1990). A One-Sentence Proof That Every Prime p Is a Sum of Two Squares. In: American Mathematical Monthly 97(2), 144.
[8]. Honsberger, R. (1970). Writing a Number as a Sum of Two Squares. In Ingenuity In Mathematics (Anneli Lax New Mathematical Library, pp. 61-66). Mathematical Association of America. doi:10.5948/UPO9780883859384.012.
[9]. Vladimirovich, D. V. and Genadievna, S. A. (2017). A generalization of fermat’s theorem on sum of two squares. Austrian Journal of Technical and Natural Sciences.
[10]. Ore, O. (1988). Number theory and its history (Dover Books on Mathematics). Dover Publications. ISBN-10: 0486656209. ISBN-13: 978-0486656205.
Cite this article
Jiang,Z. (2023). Factoring primes and sums of two squares. Theoretical and Natural Science,13,18-22.
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References
[1]. Bhaskar, J. (2008). Sum of two squares. https://www.math.uchicago.edu/~may/VIGRE/ VIGRE2008/REUPapers/Bhaskar.pdf.
[2]. Homeworkhelp. Com and Inc. Factoring and Primes (High School Math).
[3]. Stewart, I. and Tall, D. (2001). Algebraic Number Theory and Fermat's Last Theorem: Third Edition. A K Peters/CRC Press. ISBN-10: 1568811195. ISBN-13: 978-1568811192.
[4]. Edwards, H. M. (1977). Fermat's last theorem: A genetic introduction to algebraic number theory. In: Graduate Texts in Mathematics. Springer New York, NY.
[5]. Stein, M. R. and Dennis, R. K. (1989). Algebraic K-Theory and Algebraic Number Theory: Comtemporar Math., 83. American Mathematical Society, Providence.
[6]. Rosen, K. H. (2000). Elementary number theory and its applications. Addison Wesley. ISBN-10: 0201870738. ISBN-13: 978-0201870732.
[7]. Zagier, D. (1990). A One-Sentence Proof That Every Prime p Is a Sum of Two Squares. In: American Mathematical Monthly 97(2), 144.
[8]. Honsberger, R. (1970). Writing a Number as a Sum of Two Squares. In Ingenuity In Mathematics (Anneli Lax New Mathematical Library, pp. 61-66). Mathematical Association of America. doi:10.5948/UPO9780883859384.012.
[9]. Vladimirovich, D. V. and Genadievna, S. A. (2017). A generalization of fermat’s theorem on sum of two squares. Austrian Journal of Technical and Natural Sciences.
[10]. Ore, O. (1988). Number theory and its history (Dover Books on Mathematics). Dover Publications. ISBN-10: 0486656209. ISBN-13: 978-0486656205.