The History of the Formulation of Ideal Theory

Reeve Garrett November 28, 2017

1 Using complex numbers to solve Diophantine equations

From the time of Diophantus (3rd century AD) to the present, the topic of Diophantine equations (that is, polynomial equations in 2 or more variables in which only integer solutions are sought after and studied) has been considered enormously important to the progress of mathematics. In fact, in the year 1900, David Hilbert designated the construction of an algorithm to determine the existence of integer solutions to a general Diophantine equation as one of his “Millenium Problems”; in 1970, the combined work (spanning 21 years) of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson showed that no such algorithm exists. One such equation that proved to be of interest to mathematicians for centuries was the “Bachet equation”: x2 +k = y3, named after the 17th century mathematician who studied it. The general solution (for all values of k) eluded mathematicians until 1968, when Alan Baker presented the framework for constructing a general solution. However, before this full solution, Euler made some headway with some speciﬁc examples in the 18th century, speciﬁcally by the utilization of complex numbers.

Example 1.1 Consider the equation x2 + 2 = y3. (5, 3) and (−5, 3) are easy to ﬁnd solutions, but it’s 2 not obvious√ whether√ or not there are others or what√ they might be. Euler realized by factoring x + 2 as (x + 2i)(x − 2i) and then using the facts that Z[ 2i] is a UFD and√ the factors√ given are relatively prime that the solutions above are the only solutions,√ namely√ because (x + 2i)(x − 2i) being a cube forces each of these factors to be a cube (i.e. (x + 2i) = (a + b 2i)3 for some integers a and b), from which we deduce our solutions to x2 + 2 = y3 by equating coeﬃcients. Moreover, we see from the resulting system of equations that (5, 3) and (−5, 3) are the ONLY solutions.

With this insight, Euler then realized that the proof for Fermat’s Last Theorem would necessarily require similar methods. For example, as he realized in the year 1753 with x3 + y3 = z3, by considering the equation in Z[ζ3] (where ζ3 is a primitive cube root of 1), which is a UFD, one obtains a contradiction by assuming the solvability of the equation. Indeed, if there were a triple (x, y, z) satisfying this equation (z > 0), then 3 3 3 2 3 x + y = z would force (x + y)(x + ζ3y)(x + ζ3 y) = z , meaning each of the 3 factors on the left hand side 2 would have to be cubes of numbers that look like a + bζ3 + cζ3 . Then, using these, we could find a triple (a, b, c) (where 0 < c < z) solving the equation. Iterating, we get an infinite decreasing sequence of positive integers z > c > ··· , which is absurd. Fermat gave a valid proof that his last theorem holds for n = 4. It is also apparent that if n = pm where p is an odd prime, then xn + yn = zn would hold if and only if (xm)p + (ym)p = (zm)p. Therefore, it would suffice to prove Fermat’s Last Theorem for odd primes p. Therefore, we restrict to this setting, if p p p 2 p−1 p we write x + y = z as (x + y)(x + yζp)(x + yζp ) ··· (x + yζp ) = z , assuming Z[ζp] is a UFD, we’d get a contradiction in the same was as p = 3, thus proving Fermat’s Last Theorem. This is the purported proof that Lamépresented in 1847 which was disproved by Ernst Kummer, who showed that Z[ζ23] is NOT a UFD. In fact, Z[ζp] is NEVER a UFD if p ≥ 23. 2 3 22 Recall that the norm of an element f(ζ23) ∈ Z[ζ23] is the product N(f(ζ23)) = f(ζ23)f(ζ23)f(ζ23) ··· f(ζ23 ), which is an ordinary integer. With the help of Jacobi in this work, he found that the norm of any element 2 2 f(ζ23) ∈ Z[ζ23] is an integer of the form (x + 23y )/4, and this means that no such f(ζ23) can have norm 47 because 4·47 = 188 can’t be written in the form x2 +23y2 where x and y are integers. This led him to realize 21 21 that in Z[ζ23], since N(1 − ζ23 + ζ23 ) = 47 · 139, 1 − ζ23 + ζ23 is irreducible (if you could factor it, something

1 would have norm 47) and not prime (it divides 47 · 139 but can’t divide 47 or 39 because it doesn’t divide 22 139 21 their norms, 47 and 47 ). Thus, we’ve written 47 · 39 = N(1 − ζ23 + ζ23 ) as a product of 22 irreducible 23−n −n 10 13 8 15 7 16 2 elements. Also, notice that since ζ23 = ζ23 , we have that N(ζ23 + ζ23 + ζ23 + ζ23 + ζ23 + ζ23 ) = 47 10 13 8 15 4 19 2 is a product of 11 irreducible elements, each squared, and N(ζ23 + ζ23 + ζ23 + ζ23 + ζ23 + ζ23 = 139 is a product of 11 irreducible elements, each squared. Therefore, by taking half the factors from each, we can write ±47 · 139 as a product of 22 irreducible factors in a DIFFERENT way than we did before. At the time, Kummer had been studying higher reciprocity laws and was seeing the connection between these and the study of cyclotomic ﬁeld extensions. One might ask what reciprocity laws have to do with cyclotomic extensions. Before we give a sense of Kummer’s work on that question, it’s important to consider the context: namely the ﬁndings and methods of Gauss when working with quadratic forms. In his revolutionary number theory text Disquisitiones Arithmeticae (published in 1798, when he was only 21 years old; the book was revolutionary in its rigor and in how it set the foundations for number theory to come), Gauss proved the law of quadratic reciprocity and various results about binary quadratic forms (that is, polynomials of the form f(x, y) = ax2 + bxy + cy2 ∈ Z[x, y]) by introducing the Gaussian integers Z[i], and he prophetically proclaimed that ordinary arithmetic with natural numbers cannot be used to establish a general theory for higher reciprocity laws and that “such a theory demands that the domain of higher arithmetic be endlessly enlarged.” Now, let’s return to Kummer. Recall that the Legendre symbol for two distinct odd primes p and q p is denoted by q and is 1 if p is a square modulo q and −1 if not. The law of quadratic reciprocity states that

 q p  if p ≡ 1 or q ≡ 1 mod 4 = p q q − p otherwise or equivalently

p q = (−1)(p−1)(q−1)/4. q p ∼ √ Let q ≡ 1 mod 4. In general, Gal( (ζq)/ ) = q−1 and (ζq) contains ( q). Have σp denote the √ Q Q Z Q Q σp( q) √ q p √ p−1 (p−1)/2 Frobenius map a 7→ a . q = ( q) ≡ q mod p, so this is 1 if and only if p = 1 by Euler’s a (p−1)/2 criterion, which states p ≡ a mod p for any prime p and any natural number a coprime to p. √ The case q ≡ 3 mod 4 is solved similarly by considering Q( −q) instead, which gets us (−1)(p−1)/2 instead. With ideas like this in mind, and with the formulation of his “ideal numbers,” Kummer was able to extend this kind of reasoning to higher reciprocity laws (i.e. determining when p ≡ an mod q for some integer a).

2 Kummer’s Ideal Numbers in the Cyclotomic Integers Case

21 Let’s return to the case of Z[ζ23] and f(ζ23) = 1−ζ23 +ζ23 . What I didn’t mention is that we CAN’T ﬁnd any prime factors for 47 within Z[ζ23]! Indeed, if some g(ζ23) were a prime factor of 47, its norm would have to 22 divide N(47) = 47 AND g(ζ23) would have to divide 47·139 = N(f(ζ23)), which would mean g(ζ23) divides one of the 22 factors of N(f(ζ23)), say h(ζ23) since it’s prime, which would then mean N(g(ζ23))|N(h(ζ23)), forcing N(g(ζ23)) = 47, which is impossible! Kummer then thought: what if we introduced “ideal prime numbers” outside the given number system Z[ζ23] that could result in unique factorization into products of primes? To see how this works, let’s continue with this example. Ideally, since N(f) = 47 · 139, we would want to decompose f into a product of a prime with norm 47 and a prime with norm 139. In taking these norms, we would want no duplication and all factors in this norm to be prime, which would mean we’d decompose 47 as a product of 22 distinct ideal primes (our ideal prime and its “conjugates,” one for each multiplicand in N(f)), only one of which also divides f. N(f(ζ23)) Let P be the hypothetical prime divisor that divides both 47 and f(ζ23).Then, let Ψf (ζ23) := , i.e. f(ζ23) the product of the other multiplicands of N(f) besides f itself. Necessarily, Ψf is not divisible by P , but it

2 is divisible by all other “ideal prime factors” of 47. This means that fΨf is divisible by 47 if and only if f is divisible by P . Thus, for any m(ζ23) ∈ Z[ζ23], we say m is divisible by P if and only if N(m) is divisible n by 47 in Z23. Or, taking things one step further, we say m is divisible by P n times if and only if m · Ψm n is divisible by 47 in Z23. With years of extensive calculations in Z[ζp] for p < 23, he found that every time he tried fixing a prime q a q 6= p he could find a gq(ζp) ∈ Z[ζp] for which gq(ζp) = gq(ζp ) and N(gq) = q , where a ∈ N is minimal such a that q ≡ 1 mod p. By doing so, he noticed that N(gq) involves duplicate prime factors all appearing in a groups of a duplicates, meaning by using the factorization of q with N(gq), he obtained a factorization of q into (p − 1)/a distinct prime elements of Z[ζp] (see p.327 of [1]). He hoped to prove a general result of this sort for all prime numbers p using ideal numbers, but a means to construct such a gq no matter the setting wasn’t obvious. If he could, he would have (p − 1)/a ideal prime factors for q, and by use of the desired 0 0 0 q 0 a properties stated below, it would follow that any other gq(ζp) where gq(ζp) = gq(ζp ) and N(gq) ≡ 1 mod q (but not for a + 1) gives rise to the SAME ideal prime factors for the prime q. Since the only prime factor of p is ζp − 1, this would mean that defining ideal prime factors would be completely consistent, if he could just find a consistent way to find such gq(ζp) for each prime q! He also needed to guarantee that the ideal prime factors had the usual properties one would expect (this is how we’d show the “uniqueness” of gq as purported above): (1) each g(ζp) and h(ζp) would together have the same ideal prime factors as g(ζp)h(ζp); (2) each g(ζp) is divisible by q n times if and only if it’s divisible by each of the ideal prime factors P of q nµP times each, where µP is the number of times P divides q; (3) each g(ζp) 6= 0 should have a finite number of ideal prime factors; (4) the introduction of ideal prime factors should keep divisibility in Z[ζp] consistent; (5) each g(ζp) and h(ζp) that have the same ideal prime factors must only differ by a unit factor; (6) taking norms should be consistent with the usual norm defined in Z[ζp]. As originally published in 1847, Kummer didn’t have the construction of the gq’s (and, consequently, proofs for any of these properties) nailed down in all cases, but he managed to prove the existence of these g .. q in 1857 (in Uber die den Gaussischen Perioden der Kreisteilung etsprechenden Congruenzwurzeln). However, few noticed these defects; the kernel of truth and elegance of his theory was apparent, as was the necessity to generalize it. Still, despite establishing the existence of these gq, for some primes p, he couldn’t define gq for at least some choices of q. In fact, he stated that defining ideal factors is impossible in some cases (largely due to issues with discriminants).

3 Generalizing to Algebraic Integers

The next step one would naturally attempt is to extend the notion of ideal prime factors to Z[α], where α is the solution to any polynomial equation in Z[x]. In 1858, Leopold Kronecker (a former student of Kummer) claimed he had shown that Kummer’s “ideal prime” theory could be extended “without diﬃculty” to this setting; however, this was erroneous, as Richard Dedekind pointed out, and this could only be ﬁxed by considering a larger ring inside Q(α) containing Z[α]. Let’s see this with an example. √ √ √ Example 3.1 Let α = −3. Then, Z[α] = {a + b −3 : a, b ∈ Z}. Notice that (1 + −3)3 = −8. This then leads us to√ notice that, since any positive integer k can be decomposed as k = 3j + i (0 ≤ i ≤ 2), if k 3j+i 3 j i j i j j+1 i k we let b = 1 + −3, then 8b = 8b = 8(√b ) b = 8(−8) b = (−1) 8 b , which means 8b is always divisible by 2k. Since 2 doesn’t divide 1 + −3, Kummer’s theory would (if it generalized) guarantee an ideal prime factor P of 2 that would divide 2 µP (2) times and divide b µP (b) times, where µP (2) > µP (b). k k However, since we established 2 always divides 8b , we have kµP (2) ≤ µP (8) + kµP (b) for all k, meaning k[µP (2) − µP (b)] ≤ µP (8) for all k, so then µP (2) − µP (b) ≤ 0 and µP (2) ≤ µP (b), a contradiction!

So, we need to pass to larger rings sometimes to make factorizations into ideal primes make sense (in this √ −1+ −3 case, we pass to Z[ 2 ] and the contradiction found goes away). The larger rings we’d pass to in order to resolve these problems would then be our “integers” in Q(α)(α a root of a polynomial in Z[x])! One observation to be made is that it’s suﬃcient just to consider α as the root of a monic polynomial; we n n−1 don’t lose any generality. Indeed, if α satisﬁes the equation anx + an−1x + ··· + a1x + a0 = 0, then by

3 n−1 n n−1 n−2 n−2 1 multiplying both sides by an we’d get (anx) +an−1(anx) +(an−2an)(anx) +···+(a1an )(anx) + n−1 (a0an ) = 0. We then consider (anα) instead of α in our field and ring extensions. This still doesn’t resolve the question of what elements of Q(α) we should consider “integers” (henceforth referred to as algebraic integers). To motivate the formulation of this definition, Kronecker stated in 1882 that there should be “Begriffsbestimmung” (English translation: “conservation of the determination of concepts”), meaning: (1) sums and products of algebraic integers must be algebraic integers too; (2) a ∈ Q is an algebraic integer if and only if a ∈ Z; (3) conjugates of algebraic integers (that is, different roots of the same polynomial) should be integers. It is believed (though unfortunately not completely substantiated) that this led Dedekind and Kronecker to define an algebraic integer in Q(α) to be the root of a monic polynomial with coefficients in Z. More details about the inspiration behind this definition are unfortunately sparse. In modern terms, we would say that the ring of algebraic integers Oα in Q(α) is the integral closure of Z in Q(α). Indeed, the desired properties (1)-(3) hold in Oα! But a new pair of questions then arises: how do we find algebraic integers, and how do we determine which elements of Q(α) are algebraic integers? The key observation to be made in answering this question is that the index [Oα : Z[α]] is finite. Let’s see how Dedekind arrived at this conclusion. He first noted the following.

Lemma 3.2 For any α, you can find an integer Nα 6= 0 such that Nαβ ∈ Z[α] for all β ∈ Oα. Proof. Kronecker’s proof of this (in English) is on page 335 of [1]. It’s a consequence of a clever appli- cation of the fundamental theorem of symmetric polynomials, which states that any symmetric polynomial in X1, ..., Xn with coefficients in a ring R can be expressed as a polynomial with coefficients in R and variables given by the elementary symmetric polynomials e (X , ..., X ) = P X X ··· X k 1 n 1≤j1

Proposition 3.3 If α satisﬁes a degree n polynomial over Q, then there are n algebraic integers ω1, ..., ωn in Q(α) (called an integral basis) such that any a ∈ Oα is a Z-linear combination of these ωi. √ Finally, we have one other loose end: as we saw with a = 8, b = 1 + −3, and c = 2, Kummer’s theory cannot be generalized to any ring where there exist elements a 6= 0, b, and c such that ck | abk but c - b. It turns out that passing to Oα ﬁxes this problem, as Dedekind noted (also in 1871).

k k Theorem 3.4 Let a,b, and c in Oα be such that a 6= 0 and ab /c is in O for all k. Then, b/c ∈ O.

4 Transitioning from Kummer’s Ideal Numbers to Dedekind’s Ide- als

As we started to see in the previous section, in passing to the ring of integers Oα we manage to overcome the barriers to generalizing Kummer’s theory of ideal primes to Q(α) (where α is algebraic) and then to arbitrary finite dimensional algebraic extensions of Q (by the primitive element theorem). We call such extensions number fields. In this section, we’ll see more of the specifics that Dedekind wrote in 1871 in the second edition of Dirichlet’s Vorlesungen, but recast in a language that’s more amenable to Kummer’s theory.

Deﬁnition 4.1 (1) Let K be a number ﬁeld and OK be the integral closure of Z in K, let Ψ ∈ OK , and let p ∈ Z be prime. Then, we say (Ψ, p) represents an ideal prime factor of p if (1) Ψ 6≡ 0 mod p and (2) for any pair β ∈ OK and γ ∈ OK such that βγΨ ≡ 0 mod p, either βΨ ≡ 0 mod p or γΨ ≡ 0 mod p.

4 µ µ (2) We say β ∈ OK is divisible µ times by (Ψ, p) if βΨ ≡ 0 mod p , and we’ll write this as β ≡ 0 mod (Ψ, p)µ. (3) If (Ψ0, p) and (Ψ1, p) are ideal prime factors where every β divisible by one is also divisible by the other the same number of times, we say (Ψ0, p) and (Ψ1, p) represent the same ideal prime factor of p. (4) Let {ω1, ..., ωn} be an integral basis for OK over Z and let (Ψ0, p) and (Ψ1, p) be ideal prime factors, where Ψ0 = k1ω1 + ··· + knωn and Ψ1 = l1ω1 + ··· + lnωn. Then, we say Ψ0 ≡ Ψ1 mod p if ki ≡ li mod p for all i. It’s fairly clear that if Ψ0 ≡ Ψ1 mod p implies that (Ψ0, p) and (Ψ1, p) represent the same ideal prime factor of p, and we have pn congruence classes of algebraic integers mod p. (5) Given an arbitrary β ∈ OK and prime p, we say γ ≡ δ mod (β, p) if βγ ≡ βδ mod p.

Within this framework, Dedekind proved the following :

Theorem 4.2 (a). There is a well-deﬁned exact multiplicity by which any ideal prime (Ψ, p) divides an arbitrary β ∈ OK . (b). If (Ψ, p) divides β exactly ν times and divides β0 exactly ν0 times, then (Ψ, p) divides ββ0 exactly ν + ν0 times. (c). If β ∈ OK is divisible by all ideal prime factors of p with multiplicity at least as great as the multiplicity with which they divide p, then β is divisible by p. (This here is perhaps the central theorem in his work published in 1871). The same can be said for any general γ ∈ OK in place of p. (d). For any ideal prime (Ψ, p), p is divisible by (Ψ, p) exactly one more time than Ψ is. (e). Any β ∈ OK is divisible by a ﬁnite number of ideal prime factors. (f). If ≡ mod (β, p) is prime, meaning γδ ≡ 0 mod (β, p) implies either γ ≡ 0 mod (β, p) or δ ≡ 0 mod (β, p), then (β, p) is an ideal prime factor of p. (g). If ≡ mod (β, p) is NOT prime, then for some γ ∈ OK , ≡ mod (βγ, p) is prime.

5 Deﬁning Ideals

What was done in the immediately preceding section was not satisfactory to Dedekind, however. Products of ideal prime factors might not be in OK , and it’s hard to make sense of what they would actually be in that case. Ideal numbers are vaguely defined and dependent on both explicit representations of factors and arbitrary choices of Ψ, which is defined solely from divisibility tests. Dedekind wanted a specific definition that was based on intrinsic properties of the factors, rather than arbitrary choices. Toward that end, he noticed that an “ideal complex number” is defined completely by the actual algebraic integers it divides, which led him to the following definition.

Deﬁnition 5.1 Given the formal product of ideal prime factors (Ψ1, p1) ··· (Ψk, pk), which we call an ideal complex number, the ideal deﬁned by that product is the set of all α ∈ OK this product divides.

Of course, it’s apparent that the ideal deﬁned by a complex product is an ideal in the sense of our modern terminology (it’s important to note that it may not necessarily be principal since products of ideal prime factors may give rise to algebraic integers that have a common factor but don’t completely coincide up to unit multiplication). Moreover, it turns out that these correspond completely to arbitrary ideals (in the sense of our modern terminology) in the ring OK !

Theorem 5.2 The natural mapping deﬁned above from ideal complex numbers to nonzero ideals (in modern terminology) is one-to-one and onto. Moreover, any ideal may be written as a ﬁnite product of prime ideals uniquely.

Proof. (Sketch) Let I be an ideal (in the modern sense) and β ∈ OK \ I. Given σ and τ in OK , we’ll say σ ≡ τ mod (β, I) if β(σ − τ) ∈ I. We’ll be using congruence mod (β, I) to build a prime ideal containing I. Since some k ∈ Z must be in I (because any β ∈ OK must have an integer k that divides it, namely the constant term of the monic polynomial in Z[x] it satisfies), the number of congruence classes mod I is finite (this is most apparent when we think in terms of the integral basis for OK ) and therefore the number of congruence classes mod (β, I) must be finite (because these are simply the elements in annOK /I (β)). Dedekind then shows that for some product of integers γ1 ··· γl that P = {δ : δ ≡ 0 mod (βγ1 ··· γl,I)} is

5 a prime ideal. [Note: Edwards seems to have made an error in this “whittling down” that I’m not sure how to fix.] Since OK is of Krull dimension 1 (meaning all nonzero prime ideals are also maximal) and P must contain some k ∈ Z and therefore some prime p ∈ Z dividing k (since congruence mod P is prime), P must be equal to one of the prime ideals Pi corresponding to the ideal prime factors (Ψi, p) of p. Moreover, (Ψ, p) divides all elements of I with multiplicity greater than that with which it divides β. Let’s return to our general I 6= 0. I contains some k ∈ Z \{0}. Using the integral basis, pick a complete set of representatives of A mod k, say {σ1, ..., σm}. Now, single out the ideal prime factors (Ψi, pi) dividing k and let µi be the maximal multiplicity with which (Ψi, pi) divides either k or one of the σi’s (pick the µi biggest). Then, A is in fact the ideal corresponding to the ideal complex number Π(Ψi, pi) . Note: We used (Ψ, p) to define a prime ideal, but in general (as Dedekind did), we could define a simple 2 prime ideal to be a prime ideal P for which there is a pair (ν, µ) ∈ OK such that P consists of all σ ∈ OK such that νσ ≡ 0 mod µ. It turns out that we may just restrict to simple primes (Ψ, p) like we did without loss of generality and every prime ideal is a simple prime ideal. Moreover, Dedekind showed that an ideal A contains B if and only if B = AC for some ideal C, so it makes sense in this case to say A divides B. When we refer to the product ideal AC we mean Pm AC = { i=0 aici : ai ∈ A, bi ∈ B, m ∈ N}. Let’s see how Dedekind’s approach applies to complete Kummer’s approach in a concrete setting. In the next example, we’ll abide by the notation that (a, b) = {ra + sb : r, s ∈ OK } is the ideal generated by a and b. √ √ Example 5.3 Consider the domain Z[ −5]. This is in fact THE ring of integers√ in Q( √−5) because −5 6≡ 1 mod 4, and it is not a unique factorization domain because 6 = 2 × 3 = (1√ + 5i)(1 − 5i). In Kummer’s setting, we would want√ an ideal prime factor that divides both 2 and 1 + 5i, an√ ideal prime factor that divides both 3 and 1 + √5i, and√ an ideal√ prime factor√ that divides both 3 and 1 − 5i; as we’ll see, these ideal prime factors are 2, (1 +√ 5i)/ 2, and (1 − 5i)/2, respectively. In Dedekind’s setting, we define the ideal prime√ to be P √= (1 + 5i, 2), and√ passing√ to modern notation, this is a prime ideal. Moreover, 2 2 P = ((1 + 5i) , 2(1 + √5i), 4) = (−4 + 2 5i, 2 + 2 5i, 4) = (2), the principal√ ideal generated√ by 2, so we may indeed think√ of P as 2 in a sense. Similarly,√ if we set Q = (3, 1 + 5i) and R = (3, 1 − 5i), we see PQ = (1 + 5i), QR = (3), and PR = (1 − 5i). Again√ going√ from prime ideals to ideal prime factors, we think of√ Q as√ P Q/P , so we’d intuitively get Q ∼ (1 + 5i)/ 2; similarly with R thought of as P R/P , R ∼ (1 − 5i)/ 2. These are all the ideal prime factors we need since " √ # " √ # √ √ 1 + 5i 1 − 5i 6 = 2 × 3 = [( 2) × ( 2)] × √ × √ and 2 2 " √ # " √ # √ √ √ 1 + 5i √ 1 − 5i 6 = (1 + 5i) × (1 − 5i) = 2 × √ × 2 × √ , 2 2 and both of these factorizations are the same, meaning we have a unique factorization for 6, as desired. In fact, unique factorization (with prime ideals) is completely restored in this setting, as with all rings of integers in algebraic number fields!

References

[1] H.M. Edwards, The Genesis of Ideal Theory Archive for History of Exact Sciences, Volume 23, Springer Verlag (1980), p. 322-378. [2] I. Kleiner, A History of Abstract Algebra. Birkhauser (2007). Relevant material: p. 48-54. [3] I. Kleiner, From Numbers to Rings: The Early History of Ring Theory. Elem. Math, 53 (1998), p. 18-35. [4] J. J. O’Connor and E.F. Robertson, The Development of Ring Theory. Self-published notes: http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Ring theory.html [5] P. Stevenhagen, Kummer Theory and Reciprocity Laws, Self-published notes: http://websites.math.leidenuniv.nl/algebra/Stevenhagen-Kummer.pdf