Ferdinand Gotthold Max Eisenstein

Ferdinand Gotthold Max Eisenstein was born in Berlin on April 16, 1823. His family was not well off. His father had served in the Prussian army for 8 years and when he returned to civilian life, he moved from the job to job throughout Ferdinand's youth.

From the earliest age, he suffered from bad health as did all his brothers and sisters. He was the only one of the six children to survive past childhood. All of his brothers and sisters died at an early age of meningitis. Eisenstein also suffered through meningitis but was somehow able to survive.

From an early age, he showed a great talent for music. He was able to play the piano and composed songs throughout his short life. When he was 11 years old, his parents sent him to a military academy near Berlin. He hated the discipline and routines and yet it was here that he developed his love for mathematics. One of his teachers taught mathematics by presenting theorems and then asking each student to work alone on the proof. Once a student proved the theorem, the teacher presented the next theorem and asked the student to likewise prove this one. Eisenstein found this approach very enjoyable and had proved 100 elementary theorems in the time that students were expected to solve 11 or 12.

In 1837, at the age of 14, he entered Gymnasium in Berlin where his mathematical talents were recognized by his teachers. By the age of 15, he was going beyond the mathematical curriculum and started studying mathematical books on his own.

In 1842, he purchased a copy of Gauss's Disquisitiones arithmeticae and became fascinated with number theory. When his father moved to England, he was able to meet with Hamilton and in this way was introduced to the ideas of Abel another very famous mathematician who died young.

In 1843, he enrolled at the University of Berlin. During his first year, he was releasing new math articles almost weekly. In 1844, Crelle's Journal published 27 articles, 16 of which were from Eisenstein. In March 1844, Eisenstein met von Humboldt who would become a big sponsor of his. Eisenstein hated to receive grants that were given grudgingly. Through out Eisenstein's life, von Humboldt worked hard to help him get position and funds.

In June 1844, Eisenstein went on a trip to meet Gauss. Gauss had a reputation for being very hard to impress. Eisenstein had sent Gauss some of his papers and Gauss was very impressed. Eisenstein had generalized many of Gauss's results.

In February 1845, Eisenstein received an honorary doctorate thanks to the help of Kummer and Jacobi. He would later be involved with a priority dispute with Jacobi.

In 1847, Eisenstein began to lecture at the University of Berlin. Reimann attended one of these lectures and there is speculation that Eisenstein was very influential on Reimann's famous paper on the Zeta function.

In 1848, there was political unrest in Germany and shots were fired on German soldiers from a house while Eisenstein was there. Eisenstein was arrested. He was later released but his arrest made it more difficult for him to get funds.

Despite these setbacks, Eisenstein continued to publish. In 1851, he was elected to the Gottingen Academy and in 1852, he was elected to the Berlin Academy.

Eisenstein died on October 11, 1852 of pulmonary turbucolosis at the age of 29.

Eisenstein made significant contributions in quadratic forms, generalizing the results of Gauss, higher reciprocity laws that generalized Gauss's quadratic reciprocity result.

Gauss would say that the three most brilliant mathematicians of all time were Archimedes, Newton, and Eisenstein.

The details for this blog were taken from the following sources:

• MacTutor biography of Eisenstein
• Translation of Eisenstein's Autobiography
  

Euclidean Integers

A quadratic integer is considered Euclidean if it can be characterized by a division algorithm. This is the algorithm which states that division of any integer by an nonzero integer results in two unique values: a quotient and a remainder and the norm of the remainder is always less than the norm of the divisor.

In a previous blog, I gave the proof for this with regard to Gaussian Integers and with regard to rational integers. So, using these proofs, I have shown that both Gaussian Integers and rational integers are Euclidean.

Since the greatest common divisors algorithm from Euclid derives from the division algorithm, we can also conclude that all Euclidean integers have a greatest common denominator that is a linear combination of two other integers. It is also straight forward to show that all Euclidean integers are characterized by unique factorization.

This means that one sure path to establishing unique factorization is to show that a quadratic integer is Euclidean. Interestingly, it turns out that there are quadratic integers which are not Euclidan which still possess unique factorization.

Definition of Euclidean Integer: A quadratic integer a + b√d is Euclidean if for all integers α, β where β is nonzero, there exists a unique value δ such that δ = α - η*β and absolute(Norm(δ)) is less than absolute(Norm(β)).

Lemma: Z[√2], Z[√-2], Z[√3] are Euclidean

(1) For purposes of this proof, assume that √d = √2, √-2, or √3.

(2) There exists a,b,e,f such that [from the definition quadratic integers]
α = a + b√d
β = e + f√d

(3) α/β = (a + b√d)/(e + f√d) = (a + b√d)(e - f√d) /(e² - f²d)=
(ae - af√d + be√d - bdf)/(e² - f²d) =
(ae - bdf)/(e² - f²d) + √d(be - af)/(e² - f²d)

(4) Let r = (ae - bdf)/(e² - f²d), s = (be - af)/(e² - f²d) where r,s are rational but not necessarily integer. [For a review of rational numbers and their properties, see here]

(5) So α/β = r + s√d.

(6) We know that there exists m,n which are rational integers such that:
absolute(r - m) ≤ (1/2) and absolute(s - n) ≤ (1/2). [See here for proof]

(7) Let η = m + n√d, let δ = α - β * η where η, δ are quadratic integers of type Z[√d]. [For a review of quadratic integers, see here.]

NOTE: The keypoint point is to prove that absolute(Norm(δ)) is less than absolute(Norm(β)).

(8) Norm(δ) = Norm(α - β * η) = Norm(β[(α / β) - η) =
Norm(β) * Norm([α/β] - η)

(9) This means that the key point is to prove that absolute(Norm([α/β] - η)) is less than 1.

(10) We know that:
Norm([α/β] - η) = Norm([r + s√d] - [m + n√d]) = Norm([r - m] + [s - n]√d) =
([r - m] + [s - n]√d)([r - m] - [s - n]√d) = (r - m)² - d(s - n)²

(11) Applying step(6),

gives us for d=2:
absolute((r - m)² - d(s - n)²) ≤ absolute((1/2)² - 2(1/2)²) =
absolute((1/4) - 2(1/4)) = absolute(-1/4) = 1/4 which is less than 1.

gives us for d=-2:
absolute((r - m)² - d(s - n)²) ≤ absolute((1/2)² + 2(1/2)²) =
absolute((1/4) + 2(1/4)) = absolute(3/4) = 3/4 which is less than 1.

gives us for d=3:
absolute((r - m)² - d(s - n)²) ≤ absolute((1/2)² - 3(1/2)²) =
absolute((1/4) - 3(1/4)) = absolute(-2/4) = 1/2 which is less than 1.

QED

Lemma: Z[(1+√-3)/2], Z[(1+√-7)/2], Z[(1+√-11)/2], Z[(1+√5)/2] are Euclidean

(1) For purposes of this proof, assume that √d = √-3, √-7 ,√-11 , or √5.

(2) There exists a,b,e,f such that
α = a + b√d
β = e + f√d

(3) α/β = (a + b√d)/(e + f√d) = (a + b√d)(e - f√d) /(e² - f²d)=
(ae - af√d + be√d - bdf)/(e² - f²d) =
(ae - bdf)/(e² - f²d) + √d(be - af)/(e² - f²d)

(4) Let r = (ae - bdf)/(e² - f²d), s = (be - af)/(e² - f²d) where r,s are rational but not necessarily integer. [For a review of rational numbers and their properties, see here]

(5) So α/β = r + s√d.

(6) We also know that there exists p which is a rational integer such that:
absolute(s - p/2) ≤ 1/4. [See here for proof]

(7) We also know that there exists o which is a rational integer such that:
absolute(r - p/2 - o) ≤ 1/2. [See here for proof]

(8) let η = r + s[(1 + √d)/2] which is an integer as explained in a previous blog since in all above cases d ≡ 1 (mod 4).

(9) Let δ = β*([r - p/2 - o] + [s - p/2]√d)

(10) Norm(δ) = Norm(β) * Norm([r - p/2 - o] + [s - p/2]√d)

(11) Once again, to finish this proof, we need only to show that abs(Norm([r - p/2 - o] + [s - p/2]√d)) is less than 1.

(12) Norm([r - p/2 - o] + [s - p/2]√d) = ([r - p/2 - o] + [s - p/2]√d)([r - p/2 - o] -[s - p/2]√d)
= [r - p/2 - o]² - d[s - p/2]² ≤ (1/2)² - d(1/4)² = (1/4) - d/16.

QED

In addition, here are more interesting facts:

• There are exactly 21 types of quadratic integers that are Euclidean: a quadratic integer is Euclidean if d = -11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37,41, 57, and 73.

• Z[(-1 + √-19)/2] is not Euclidean but still has unique factorization

• Z[√-5] does not have unique factorization.

The content of this blog is based in a large part on Harold M. Stark's An Introduction to Number Theory.