1: Unique Factorisation of Ideals

The next few posts will be me detailing some interesting results in the area of Maths that I hope to specialise in: Algebraic Number Theory. The first result will be the unique prime factorisation of ideals. But first, what is an ideal?

Ideals

If you’re not familiar with the definition of a ring click here, as we’ll need this for the following discussion.

An ideal, I, is a subset of a ring R if :

  • It is closed under addition and has additive inverses (i.e. it is an additive subgroup of (R, +. 0);
  • If aI and bR, then a · bI.

I is a proper ideal if does not equal R.

Ring of Integers

In order to prove the result, I need to introduce the concept of number fields and the ring of integers. A number field is a finite field extension over the rationals.

Field Extension: a field extension F of a field E  (written F/E) is such that the operations of E are those of restricted to E, i.e. is a subfield of F.

Given such a field extension, F is a vector space over E, and the dimension of this vector space is called the degree of the extension. If this degree is finite, we have a finite field extension.

So if is a number field, E would be the rationals.

Suppose is a number field. An algebraic integer is simply a fancy name for an element of F such that there exists a polynomial f with integer coefficients, which is monic (the coefficient of the largest power of x is 1), such that f(a) = 0.

The algebraic integers in a number field F form a ring, called the ring of integers, which we denote as OF. It turns out that the ring of integers is very important in the study of number fields.

Prime Ideals

If P is a prime ideal in a ring R, then for all x, y in R, if xyP, then x ∈ P or y ∈ P. As OF is a ring we can consider prime ideals in OF.

Division = Containment

We want to try and deal with ideals in the same way we deal with numbers, as ideals are easier to deal with (ideals are a sort of abstraction of the concept of numbers). After formalising what it means to be an ideal and proving certain properties of ideals, we can prove that given two ideals I and J, I dividing (written I|J) is equivalent to J containing I.

Three Key Results

Now, there are three results that we will need in order to prove the prime factorisation of ideals that I will simply state:

  1. All prime ideals P in Oare maximal (in other words, there are no other ideals contained between P and R). Furthermore, the converse also holds: all maximal ideals in OF are prime.
  2. Analogously to numbers (elements of a number field F), if I, J are ideals in Owith J|I, there exists an ideal contained in I, such that I = JK.
  3. For prime ideal P, and ideals I,J of  OF , PIJ implies PI or PJ.

Main Theorem

Theorem: Let I be a non-zero ideal in OF . Then I can be written uniquely as a product of prime ideals.

Proof:  There are two things we have to prove: the existence of such a factorisation and then its uniqueness.

Existence: If is prime then we are done, so suppose it isn’t. Then it is not maximal (by 1) so there is some ideal J, properly contained in I. So J|I, so (by 2) there is an ideal K, contained in I, such that I = JK. We can continue factoring this way and this must stop eventually (for the curious, I make the technical note that this must stop as we have an infinite chain of strictly ascending ideals, and OF is Noetherian).

Uniqueness: If P1 · · · Pr = Q1 · · · Qs, with Pi , Qj prime, then we know P1 | Q1 · · · Qs, which implies P1 | Qi for some i (by 3), and without loss of generality i = 1. So Q1 is contained in P1. But Q1 is prime and hence maximal (by 1). So P1 = Q1. Simplifying we get P2 · · · Pr = Q2 · · · Qs. Repeating this we get r = s and Pi = Qi for all i (after renumbering if necessary).

Why is this important?

For numbers, we only get unique prime factorisation in what is called a unique factorisation domain (UFD). Examples of UFDs are the complex numbers and the rationals. However, the integers mod 10 no longer form a UFD because, for example, 2*2 = 4 = 7*2 (mod 10).

However, we have the unique prime factorisation of ideals in the ring of algebraic integers of any number field. This means that we can prove many cool results by using this unique prime factorisation, which we can then translate into results about numbers in that number field. I will detail some of these in future blog posts.

M x

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Towers of Hanoi

The Problem

The Tower of Hanoi is a famous problem first posed in 1883 by E. Lucas. There are 3 rods and n disks resting on one rod, sorted in size like a pyramid with the smallest disk on top. Given that you can only move the disks one at a time and that you can never place a bigger disk on a smaller disk, the aim is to move all the disks from the left hand post to the right hand post in the smallest number of moves possible.

Towers Of Hanoi
Source: http://www.codenuclear.com

Obtaining a Recurrence

Consider n = 0, the simplest case with no discs. As there are no discs to move, we cannot make any moves, so the number of steps required is 0. Letting Sn be the number of moves required for n disks, we get S= 0.

Now, we must consider how the problem scales. With n = 1, a single step will solve the problem. With n = 2, the answer is 3 steps: one to move the top (small) disk to another rod, one step to move the big disk to the destination rod, and lastly one step to move the small disk on top of the big disk.

We now consider n discs. We need Sn-1 steps to move all disks except the biggest one, then move the biggest disks then move the sub-tower on top of that disc with Sn-1 steps again. So we have the following upper bound:

Screenshot 2019-07-03 at 5.29.06 PM.pngWhat about a lower bound? At some point we must move the biggest disk to the destination rod. To get to the biggest disk, we must have moved all disks on top of it to another rod (the sub-tower), and after having moved the biggest disk, we must move this sub-tower back on top of that rod back onto the biggest disk. Due to these constraints due to the rules of the problem, we know that for n > 0 we must take at least 2*(Sn-1) + 1 steps.

Hence, this recurrence relation gives us the exact number moves needed.

Simplifying the Recurrence

Claim: Sn = 2n – 1 for all natural n.

Proof (by Induction):

Clearly true for n = 0.

Then Sk+1 = 2*(Sk + 1 = 2*(2k – 1) + 1, by the induction hypothesis.
So Sk+1 = 2*2k – 2 + 1 = 2k+1 – 1, as required.

So the claim holds by induction.

So we have found an easy formula for the number of steps needed to solve the Tower of Hanoi problem for any n!

M x

 

I’m back!

After a long year of university filled with lots of ups and downs, I am finally back! I decided to take a year out of blog post writing to focus on my final year as an maths undergraduate. A year later, I have finished my final exams and found out that I did well enough to get a spot in the masters program! Feeling refreshed and filled with one more year of maths to share, I am ready to start writing regular posts again.

Thank you for staying subscribed despite my lack of output this year and hopefully see you in many more posts to come!

M x

The woman who could help put men on Mars

Kathleen Howell is an American scientist and aerospace engineer. Her contributions to the theory of dynamical systems have been applied to spacecraft trajectory design, which led to the use of near-rectilinear halo orbit (NRHO) in various NASA space missions.

Unlike an ordinary flat orbit, an NRHO can be slightly warped. Further, it stands on end, almost perpendicular to an ordinary orbit – hence “near rectilinear”.

NASA have decided that an NRHO would be an ideal place to put the Lunar Orbital Platform-Gateway, which is a planned way station for future human flights to the Moon and eventually Mars. The plan is for the Gateway’s circuit to pass tight over the Moon’s north pole at high speed and more slowly below the south pole, because of the greater distance from the moon.

Imagine moving your hand in circles, as if washing a window, while you walk forward. Except you’re making hand circles around the moon while walking around Earth.” – Bloomberg

Although this orbit seems to be an ordinary circuit of the moon, it’s actually part of a family of orbits, centred on an empty point, called L2 (or Lagrange Point 2). Here, around 45,000 miles beyond the far side of the Moon, the gravitational forces of the Earth and the moon are in balance with the centrifugal forces on the spacecraft.

Although we are taught in school that orbits must be around something, it is quite possible to orbit around nothing, so long as that ‘nothing’ is a Lagrange point.

“It is elegant and very rich. All the forces come together to produce an unexpected path through space” – Howell

Howell’s work build on an 18th century discovery, by Euler, who theorised that for any pair of orbiting bodies, there are 3 points in space where gravitational and centrifugal forces balance precisely. In 1772, Lagrange found two more such spots. All five are now known as Lagrange points.

Image result for ;lagrange points
Source: Wikipedia

In 2017 Kathleen Howell was elected to National Academy of Engineering “for contributions in dynamical systems theory and invariant manifolds culminating in optimal interplanetary trajectories and the Interplanetary Superhighway“.

M x

NEWS: Fields Medal Winners 2018

The four winners of the Fields Medal for 2018 have been announced. The Fields Medal is awarded every 4 years at an international gathering of mathematicians and is considered the Nobel Prize for Mathematics. However, there is one key difference: recipients must be 40 years old or younger.

This years recipients were announced on Wednesday 1st August at the International Congress of Mathematicians in Rio de Janeiro. They are:

  • Caucher Birker, 40, of the University of Cambridge in England: “for his proof of the boundedness of Fano varieties and for contributions to the minimal model program.”
  • Click here for more information.
  • Allesio Figalli, 34, of the Swiss Federal Institute of Technology in Zurich: “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability.”
  • Click here for more information.
  • Akshay Venkatesh, 36, of the Institute for Advanced Study in Princeton and Stanford University in California: “for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.” Click here for more information.
  • Peter Scholze, 30 (one of the youngest recipients), of the University of Bonn in Germany: “for transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to galois representations and for the development of new cohomology theories.” Click here for more information.
1630.jpg
Source: The Guardian

M x

 

 

Influential Mathematicians: Gauss (3)

Probability and Statistics

Gauss introduced what is now known as the Gaussian distribution: he showed how probability can be represented by a bell-shaped curve, with peaks around the mean when falls off quickly towards plus or minus infinity.

File:Normal Distribution PDF.svg
Source: Wikipedia

He also created the Gaussian function: a function of the form

{\displaystyle f(x)=ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}}

for arbitrary real constants a, b and c.

Modular Arithmetic

The modern approach to modular arithmetic was developed by Gauss in his book Disquisitiones Arithmeticae, published in 1801.  This now has application in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.

Surveying

Whilst doing a surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth and started to question what the shape of space itself was. This led him to question Euclidean geometry – one of the central tenets of the whole mathematics, which premised a flat universe, rather than a curved one. He later claimed that as early as 1800 he had already started to consider types of non-Euclidean geometry (where the parallel axiom does not hold), which were consistent and free of contradiction. However, to avoid controversy, he did not publish anything in this area and left the field open to Bolyai and Lobachevsky, although he is still considered by some to be the pioneer of non-Euclidean geometry.

This survey work also fuelled Gauss’ interest in differential geometry, which uses differential calculus to study problems in geometry involving curves and surfaces. He developed what has become known as Gaussian curvature. This is an intrinsic measure of curvature that depends only on how distances are measured on the surface, not on the way it is embedded in space.

Positive, negative and zero Gaussian curvature of a shell
Source: shellbuckling.com

His achievements during these years, however, was not only limited to pure mathematics. He invented the heliotrope, which is an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey.

Image result for heliotrope gauss
Heliotrope | Source: Wikipedia

All in all, this period of time was one of the most fruitful periods of his academic life; he published over 70 papers between 1820 and 1830.

In later years, he worked with Wilhelm Weber to make measurements of the Earth’s magnetic field, and invented the first electric telegraph.

Read part 1 here and part 2 here.

Let me know what you think of this new series! M x

 

Influential Mathematicians: Gauss (2)

Read the first part of this series here.

Although Gauss made contributions in many fields of mathematics, number theory was his favourite. He said that

“mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”

A way in which Gauss revolutionised number theory was his work with complex numbers.

Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables. Although imaginary numbers had been used since the 16th century to solve equations that couldn’t be solved any other way, and although Euler made huge progress in this field in the 18th century, there was still no clear idea as to how imaginary numbers were connected with real numbers until early 19th century. Gauss was not the first to picture complex numbers graphically (Robert Argand produced the Argand diagram in 1806). However, Gauss was the one who popularised this idea and introduced the standard notation a + bi. Hence, the study of complex numbers received a great expansion allowing its full potential to be unleashed.

Furthermore, at the age of 22 he proved the Fundamental Theorem of Algebra which states:

Every non-constant single-variable polynomial over the complex numbers has at least one root.

This shows that the field of complex numbers is algebraically closed, unlike the real numbers.

Gauss also had a strong interest in astronomy, and was the Director of the astronomical observatory in Göttingen. When Ceres was in the process of being identifies in the late 17th century, Gauss made a prediction of its position. This prediction was very different from those of other astronomers, but when Ceres was discovered in 1801, it was almost exactly where Gauss had predicted. This was one of the first applications of the least squares approximation method, and Gauss claimed to have done the logarithmic calculations in his head.

gauss_least_squares.gif
Source: The Story of Mathematics

Part 3 coming next week!

M x