Fundamental Theorem of Algebra: At least one solution

In today's blog, I continue the proof for the Fundamental Theorem of Algebra. Today, I will show the proof that all polynomials in the complex domain have at least one root that leads to 0.

Today's proof is taken from David Antin's translation of Heinrich Dorrie's 100 Great Problems of Elementary Mathematics.

Lemma 1: If f(x) = xⁿ + a₁x^n-1 + ... + a_n-1x + a_n where a_i and x are complex numbers, then there exists at least one solution r such that f(r)=0.

Proof:

(1) Let f(x) = xⁿ + a₁x^n-1 + ... + a_n-1x + a_n

(2) Assume that for all x, f(x) ≠ 0

(3) We know that there exists a value x₀ such that w₀=f(x₀) and w₀ is the smallest absolute number. [See Lemma 2 here for proof]

(4) From step #2, we can assume that absolute(w₀) is greater than 0.

(5) We can plot the minimal point, w₀, on the plane of complex numbers (see here for more details if needed)

(6) From this point, we can define a small circle K with radius R.

(7) Now, for any point x in K, x = x₀ + ζ

In other words, complex numbers form a one-to-one mapping between the possible values for f(x) and the Cartesian coordinate system. If we use the form r(cos θ + i sin θ), then we see that r is the radius of K [See here if more information needed]

(8) Using the plane of complex numbers, we know that there exists ρ, θ such that ζ = ρ(cos θ + isin θ) (see here for review of how cos θ + isin θ can be used in this situation)

(9) In the above case, ρ = the absolute magnitude of ζ

(10) So, for any value x, there exists a value w and a value ζ such that:

w = f(x) = f(x₀ + ζ) = (x₀ + ζ)ⁿ + a₁(x₀ + ζ)^n-1 + ... + a_n

(11) From the equation in #10, we can rearrange the values to get the following:

w = f(x₀) + c₁ζ + c₂ζ² + ... + c_nζⁿ = w₀ + c₁ζ + c₂ζ² + ... + c_nζⁿ

(12) Now, it is quite possible that some of the c_i values are 0 so we can rearrange the values so the first nonzero coefficient is c and the power is v, the next is c' and the power is v', and so on where each c,v are nonzero and v is less than v' is less than v'', etc.:

w = w₀ + cζ^v + c'ζ^v' + c''ζ^v'' + ...

(13) Since we assume that w₀ is nonzero, we can divide both sides by w₀ to get:

w/w₀ = 1 + (cζ^v)/w₀ + (c'ζ^v')/w₀ + (cζ^v'')/w₀ + ...

(14) Now let us define some values to make this equation more manageable:

Let q = c/w₀

Let ξ = (c'ζv' + c''ζv'' + ...)/(cζ^v)

So that:

w/w₀ = 1 + qζ^v(1 + ζξ)

(15) Now, since q, ζ are complex numbers, we can represent them both using r(cos + isin) form (see here if more information needed)

So that there exists h, λ such that:

q = h(cos λ + i sin λ)

From step #8, there exists ρ, θ such that:

ζ = p(cos θ + i sin θ)

(16) To shorten the equation we can use:

1_λ = cos λ + i sin λ

And use:

1_θ = cos θ + i sin θ

So that we have:

q = h * 1_λ

ζ = ρ * 1_θ

(17) Using step #16, we get:

qζ^v = h*1_λ*(ρ*1_θ)^v = h*p^v*1_λ*(1_θ)^v

(18) Now, using Euler's Formula (see Lemma #1, Lemma #2 here if needed), we know that:

(1_θ)^v = 1_vθ

1_vθ*1_λ = 1_{λ + vθ}

(19) Within the circle K, we can now consider only the values of x that are associated with θ = (π - λ)/v. [Since a circle includes all values of θ between 0 and 2π, see here if needed]

(20) In this case:

1_{λ + vθ} = 1_{λ + v(π - λ)/v} = 1_{λ - λ + π} = 1_π

(21) Now 1_π = cos (π) + isin(π) = -1 + i*0 = -1. [See here if needed]

(22) So, in this case, combining step #21 with step #17:

qζ^v = h*ρ^v*(-1) = -h*ρ^v

(23) Combining step #22 with step #14:

w/w₀ = 1 - hρ^v(1 + ζξ)

(24) Now, we can set the radius of K to any value so we can constrain ζξ to be as close to 0 as we wish so that for all purposes, we have:

w/w₀ = 1 - hρ^v

[The idea here is that the radius of K was selected arbitarily in step #6, any nonzero radius r will do]


(25) Now, we can choose any value for ρ so we choose a value such that ρ is greater than 0 and less than (1/h)^(1/v).

We can do this since ρ is the magnitude of ζ (see step #9). In step #19, we constrained θ and in step #24, we constrained the maximum magnitude of the circle K.

Even with all of the above constraints, we are still left with a set of values and we can select a value of x such that the magnitude is less than the radius of K and less than (1/h)^(1/v) but still greater than 0.

(26) But then hρ^v is greater than 0 and less than h*(1/h) = 1 since:

h*[(1/h)^(1/v)]^v = 1

(27) But this means there is a value of x such that w/w₀ is greater than 0 and less than 1.

(28) But this is a contradiction because it means that w is less than w₀ which is impossible from step #3.

The reasoning here is that if w/w₀ = fraction, this implies that w = w₀ * fraction which implies that w is less than w₀

(25) So we are forced to reject our assumption in step #2.

QED

References

• Heinrich Dorrie, 100 Great Problems of Elementary Mathematics

Fundamental Theorem of Algebra: Preliminaries

Cyclotomic integers allow Fermat's Last Theorem to be factored in the following way:

xⁿ + yⁿ = (x + y)(x + αy)(x + α²y)*...*(x + α^n-1y)

In the above factorization, n is an odd prime and α is a root of unity where αⁿ = 1 and αⁱ ≠ 1 for all 1 ≤ i ≤ n-1.

To establish this refactorization, I will be using the Fundamental Theorem of Algebra.

The main idea behind the Fundamental Theorem of Algebra is that for any given polynomial of a single variable of order n, there are at least n zeros to this equation in the complex domain.

This is an easy proof to state but difficult to prove. Rene Descartes and Jean le Rond d'Alembert knew about this result but it was not until Carl Friedrich Gauss that the fundamental theorem was rigorously proved. d'Alembert thought that the existence of a minimum point for a complex equation was obvious and needed no proof. I do not find this point so obvious so I begin with its proof.

The details in today's proof are taken from B. N. Delone's article on Algebra from Mathematics: Its Content, Methods, and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev and translated by S. H. Gould.

Theorem 1: Bolzano-Weierstrass Theorem

If a rectangle contains an infinite sequence of points (z₁, z₂, ..., z_n, ... in its interior, then there exists a point z₀ such that in any arbitrarily small neighborhood of z₀, there are infinitely many points of the sequence z₁, z₂, ..., z_n, ...

Proof:

(1) Let R₁ be a rectanlge that contains an infinite sequence of points.

(2) We divide it up into 4 equal parts using two lines parallel to each of its sides.

(3) At least one of these four parts contains infinitely many points, let us label it R₂

(4) We now divide up R₂ into 4 equal parts using two lines parallel to each of its sides.

(5) One of these four parts contains infinitely many points and we label it R₃

(6) In this way, we are able to generate a sequence of nested rectangles which we can label R₁, R₂, ..., R_n

(7) We can think of each side of this rectangle representing two nested intervals that exist on the x and y axises.

(8) Using Lemma 1 here, we there exists a point on the x-axis and a point on the y-axis that each of these nested intervals have in common.

(9) But this means that there is a point within the nested rectangles such that any arbitrary small neighborhood contains an infinity of points.

QED

Theorem 2: There exists a minimum point for c₀xⁿ + c₁x^n-1 + ... + c_n.

Let f(x) = c₀xⁿ + c₁x^n-1 + ... + c_n.

There exists a value x₀ such that w₀ = c₀(x₀)ⁿ + c₁(x₀)^n-1 + ... + c_n where absolute(w₀) is the minimum value.

Proof:

(1) Let g = absolute(f(0))

(2) Let G be a number greater than g.

(3) Let R be a number such that if absolute(x) > R, then absolute(f(x)) is greater than G

(4) Now, if f(0)=0, then x₀=0 and w₀=0 so we can assume going forward that f(0) ≠ 0

(5) If f(0) is greater than 0 and all f(x) ≥ g, then x₀=0 and w₀=g so we can assume that there exists at least 1 point x' such that absolute(f(x')) is less than g.

(6) Based on a nonzero g, we can set up the follow sequence:

0, g/n, 2g/n, ..., ng/n = g

(7) We can find a value i, c_n such that c_n = (i/n)g and all values absolute(f(x)) ≥ c_n.

(8) From i, we can also find a value c_n' such that c_n' = [(i+1)/n]g and there exists at least one value x' such that absolute(f(x')) is less than c_n'

(9) We can find c_n,i,c_n' regardless of the value of n so we can let n increase to infinity.

(10) Now for all values of n, we can assume that absolute(x) ≤ R since if absolute(x) is greater than R, then absolute(f(x)) is greater than G and therefore greater than g. For purposes here, let's call these values x_n

(11) So, we only need to consider the points x_n that lie inside a rectangle of sides 2R and with its center at the origin.

(12) By Theorem 1 above, there exists a point z₀ such that every neighborhood of z₀ contains infinitely many points of the sequence z₁, z₂, ..., z_n. Let us call this point x₀

(13) For any point x, we have:

absolute(f(x)) is greater than c_n = c_n' - g/n which is greater than absolute(f(x_n)) - g/n = absolute(f(x₀)) + absolute(f(x_n)) - absolute(f(x₀)) - g/n.

(14) This inequality is true for all values of n so as n increases toward infinity, we see that difference absolute(f(x_n)) - absolute(f(x₀)) becomes arbitrarily small in absolute value with g/n.

(15) Consequently, all absolute(f(x_n)) ≥ absolute(f(x₀)) so x₀ is the minimum point.

QED

References

• A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev, Mathematics: Its Content, Methods, and Meaning.
  

Cyclotomic Integers: Factoring Fermat's Last Theorem

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

The major reason why cyclotomic integers are interesting in relation to Fermat's Last Theorem is because they enable us to factor Fermat's Last Theorem in the following way:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Below I will show how I can derive this factoring using the Fundamental Theorem of Algebra.

Lemma 1: Let α be a primitive root of unity such that n is an odd prime and αⁿ = 1, and let x,y,z be integers such that xⁿ + yⁿ = zⁿ, then:

zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) We know that xⁿ - 1 has n root from the Fundamental Theorem of Algebra.

(2) We also note that for all αⁱ where 0 ≤ i ≤ n-1, we have (αⁱ)ⁿ = 1.

NOTE: αⁿ = 1 so it is really the same as α⁰.

(3) Based on #2, the Fundamental Theorem of Algebra gives us:

xⁿ - 1 = (x - 1)*(x - α)*(x - α²)*...*(x - α^n-1)

QED

Theorem 1: if n is odd, then zⁿ = xⁿ + yⁿ = (x + y)(x + αy)(x + α²y) .... (x + α^n-1y)

Proof:

(1) aⁿ - 1 = (a - 1)*(a - α)*(a - α²)*...*(a - α^n-1) [From Lemma 1 above]

(2) Since a can be any value, let a = -x/y so that:

(-x/y)ⁿ - 1 = [(-x/y) - 1]*[(-x/y) - α]*...*[(-x/y) - α^n-1] = -(x)ⁿ/yⁿ - 1

(3) If we multiply (-y)ⁿ=-(yⁿ) to both sides, we get:

xⁿ + yⁿ = (x + y)*(x + yα)*...*(x + α^n-1y)

QED

Cyclotomic Integers: Division Algorithm

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

Today, I will continue reviewing the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

Lemma 1: Criteria for division of a cyclotomic integer by a rational integer.

A cyclotomic integer is divisible by a rational integer if and only if all of its integer coefficients are congruent modulo the rational integer.

Proof:

(1) Let f(α) be a cyclotomic integer.

(2) f(α) = a₀ + a₁α + ... a_λ-1α^λ-1 [See Lemma 1 here for details]

(3) Let d be a rational integer.

(4) It is clear that if d divides f(α), then a₀ ≡ a₁ ≡ ... ≡ a_λ-1 ≡ 0 (mod d).

(5) Assume a₀ ≡ a₁ ≡ ... ≡ a_λ-1 (mod d).

(6) Then, using Corrolary 2.1 from here, we know that we can add a_λ-1 to each of the coefficients to get:

f(α) = (a₀ - a_λ-1) + (a₁ - a_λ-1)α + ... + (a_{λ - 2} - a_{λ - 1})α^λ-2 + (a_λ-1 - a_λ-1)α^λ-1

(7) But then it is clear that d divides each of the coefficients so it divides f(α).

QED

Corollary 1.1: Division Algorithm for a cyclotomic integer f(α) by a rational integer d

if f(α) = a₀ + a₁α + ... + a_λ-1α^λ-1, the result is:

[(a₀ - a_λ-1)/d] + [(a₁ - a_λ-1)/d]α + ... + [(a_λ-2 - a_λ-1)/d]α^λ-2

Proof:

(1) If d divides f(α), then a₀ ≡ a₁ ≡ ... ≡ a_λ-1 (mod d) [From Lemma 1 above]

(2) By Corollary 2.1 here, we can add constant -c to each coefficient and still maintain the same value so that:

f(α) (a₀ - a_λ-1) + (a₁ - a_λ-1)α + ... + (a_{λ - 2} - a_{λ - 1})α^λ-2 + (a_λ-1 - a_λ-1)α^λ-1

(3) From this we know that d divides each coefficient and the result of this corrollary follows.

QED

Lemma 2: Division Algorithm for Cyclotomic Integers

A cyclotomic integer h(α) is divisible by another cyclotomic integer f(α) if and only if:

h(α)*f(α)^-1 is divisible by the Nf(α)


(1) Let f(α),h(α) be cyclotomic integers.

(2) We can assume f(α) ≠ 0

(a) Assume f(α) = 0.

(b) Then, g(α) exists only if h(α) = 0 in which case g(α) can take any value.

(c) This resolves f(α) = 0 so we only need to address the case where f(α) ≠ 0.

(3) Assume f(α) * g(α) = h(α)

(4) Then, Nf(α)*g(α) = h(α) *f(α²)*f(α³)*...*f(α^λ-1) = h(α)*f(α)^-1

(5) We see that f(α) divides h(α) if and only if Nf(α) divides h(α)*f(α³)*...*f(α^λ-1).

(6) But Nf(α) is a rational integer. [See Lemma 5 here]

(7) Let i(α) = h(α)*f(α)^-1

(8) i(α) = b₀ + b₁α + ... + b_λ-1α^λ-1

(9) Using Lemma 1 above, we see that Nf(α) divides i(α) if and only if for any two coefficients b_j ≡ b_k (mod Nf(α))

(10) But from step #5, we have that f(α) divides h(α) if and only if after computing i(α) from step #8, we find that all coeffients of i(α) are congruent modulo Nf(α).

QED

Corollary 2.1: Method for determining result of division between two cyclotomic integers.

if f(α) = a₀ + a₁α + ... + a_λ-1α^λ-1 and:

h(α)f(α)^-1 = b₀ + b₁α + ... + b_λ-1α^λ-1,

the result is:

[(b₀ - b_λ-1)/Nf(α)] + [(b₁ - b_λ-1)/Nf(α)]α + ... + [(b_λ-2 - b_λ-1)/Nf(α)]α^λ-2

Proof:

(1) Let f(α), g(α), h(α) be cyclotomic integers with h(α) = f(α)g(α).

(2) Multiplying both sides by f(α)^-1 gives us:

Nf(α)g(α) = h(α)f(α)^-1

(3) Since Nf(α) is a rational integer, we can apply Corollary 1.1 above to get the desired result.

QED

Cyclotomic Integers: Units and Primes

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

Today, I will continue reviewing the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

1. Unit

Definition 1: Cyclotomic Unit

A cyclotomic unit is a cyclotomic integer whose norm is 1.

So, if f(α) is a unit, then f(α)*f(α²)*...*f(α^λ-1) = 1.

Definition 2: Cyclotomic Inverse

If f(α) is a unit, then f(α²)*..*f(α^λ-1) is called the inverse and it represented as f(α)^-1.

Lemma 1: if f(α)*g(α) = 1, then f(α) is a unit and g(α) = f(α²)*f(α³)*...*f(α^λ-1).

Proof:

(1) Let f(α)*g(α) = 1.

(2) Nf(α)*Ng(α) = N(1) = 1 [See Lemma 6, here]

(3) So, Nf(α) = 1 [See Lemma 5, here]

(4) So, Nf(α) = f(α)*f(α²)*...*f(α^λ-1) = 1 [Definition of Norm for Cyclotomic integers, here]

(5) So, f(α)*f(α²)*...*f(α^λ-1) = f(α)*g(α) [Combining step #1 with step #4]

(6) Dividing both sides of step #5 by f(α) gives us the desired result.

QED

Lemma 2: A cyclotomic unit is a factor of any cyclotomic integer.

Proof:

(1) Let f(α) be a unit.

(2) Let h(α) be any cyclotomic integer.

(3) Let g(α) = h(α)*f(α)^-1 [See definition 2 above]

(4) Then, h(α) = g(α)*f(α)

QED

2. Cyclotomic Primes

Definition 3: Irreducible

A cyclotomic integer h(α) is irreducible if for any factorization h(α)=f(α)*g(α), either f(α) or g(α) is a unit.

Definition 4: Cyclotomic Prime

A cyclotomic integer h(α) is prime if:

(a) if h(α) divides f(α)*g(α), then h(α) divides f(α) or g(α).

(b) there exists at least one cyclotomic integer f(α) that h(α) does not divide.

(c) if h(α) is not a factor of f(α) and it is not a factor of g(α), then it is not a factor of f(α)*g(α)

In rational integers, all irreducible nonunits are also primes. One of the question that needs to be addressed is whether this is still the case with cyclotomic integers. I will answer this question in a later blog.

3. More Properties of Units

Lemma 3: a unit * a unit = a unit

Proof:

(1) Let g(α),h(α) be units.

(2) By Lemma 6, here, if i(α)=g(α)*h(α), then Ni(α) = Ng(α)*Nh(α) = 1*1 = 1.

QED

Lemma 4: 1/unit = a unit

Proof:

(1) Let h(α) be a unit

(2) Nh(α) = h(α)*h(α²)*...*h(α^λ-1) = 1. [See Definition 2, here for definition of norm for cyclotomic integers]

(3) From (#2), we can see that:

1/h(α) = h(α²)*...*h(α^λ-1)

(4) Further, we can see that:

Norm(1/h(α)) = Nh(α²)*N(α³)*...*Nh(α^λ-1) = 1*1*1*...*1 = 1 [See Lemma 6, here]

QED

Lemma 5: unit/unit = unit

Proof:

(1) Let h(α),g(α) be units.

(2) h(α)/g(α) = h(α)*(1/g(α))

(3) From Lemma 4 above, we know that (1/g(α)) is a unit.

(4) From Lemma 3 above, we know that a unit*unit = unit.

QED

Basic Properties of Cyclotomic Integers

Today's blog continues the discussion of Kummer's proof of Fermat's Last Theorem for regular primes. If you would like to review the historical context for this proof, start here.

Today, I will review the basic properties of cyclotomic integers. Today's content comes directly from Chapter 4 of Harold M. Edwards' Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory.

1. Notation

For Kummer's notation, he used λ to represent the odd prime number and α to represent the root of unity so that we have:

Definition 1:
α^λ = 1

2. Standard Form of Cyclotomic Integers

Lemma 1:
If a₀, a₁, ... a_λ-1 are integers, then all cyclotomic integers for a given value of λ can be represented in the following form:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

Proof:

(1) Let's assume that we have cyclotomic integer = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + a_λα^λ

(2) By definition 1 above, α^λ = 1

(3) So that we have:

(a₀ + a_λ) + a₁α + a₂α² + ... + a_λ-1α^λ-1

(4) We can do the same thing for any power of αⁱ where i ≥ λ

(5) So we can conclude that all values can be reduced to the form required.

QED

Lemma 2: For any given value of λ, 1 + α+α² + ... + α^λ-1 = 0

Proof:

(1) Since α^λ = 1, we have:

1 + α+α² + ... + α^λ-1 =α^λ + α+α² + ... + α^λ-1 =

= α(α^λ-1 + 1 + α+α² + ... + α^λ-2)

(2) Now, we know that α ≠ 0 since 0^λ = 0 which contradicts with definition 1.

(3) We also know that α ≠ 1 since α is a λth root of unity [using Euler's Identity, see here], we know that α = e^2iπ/λ

(4) So, therefore, 1 + α+α² + ... + α^λ-1 = 0

QED

Corollary 2.1: for any given integer c, a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 = (a₀ + c) + (a₁ + c)α + (a₂ + c)α² + ... + (a_λ-1 + c)α^λ-1.

Proof:

(1) 1 + α + α² + ... + α^λ-1 = 0 [From Lemma 2 above]

(2) c + cα + cα² + ... + cα^λ-1 = c*0 = 0

(3) So that:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + 0 =

= a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1 + c + cα + cα² + ... + cα^λ-1 =

= (a₀ + c) + (a₁ + c)α + (a₂ + c)α² + ... + (a_λ-1 + c)α^λ-1.

QED


3. Conjugates

Since each cyclotomic value can be represented as:

a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

Kummer used the following shorthand to represent a cyclotomic integer:

f(α), g(α), φ(α), F(α), etc.

One important point that we find is that if f(α) = g(α), then f(α²) = g(α²) and so on up until λ - 1.

Lemma 2.5: Conjugates preserve relations between equations

That is, if f(α) = g(α), then f(αⁱ) = g(αⁱ) where i is a positive number less than λ, αⁱ ≠ 1 and α^λ = 1.

Proof:

(1) Let f(α) = a₀ + a₁α + ... + a_λ-1α^λ-1

(2) For any value f(αⁱ) we see that:

f(αⁱ) = a₀ + a₁αⁱ + ... + a_λ-1α^i*(λ-1)

(3) In step #1, let j be the possible values ranging from 1 to λ -1. Combining this with step #2, we get:

f(αⁱ) = ∑ a_jα^j*i

(4) To prove this lemma, we need to show each element j*i is congruent to a unique value of i modulo λ

In other words, we are trying to prove that each element of the f(αⁱ) is distinct.

(5) This turns out to be the case from Lemma 1 here.

QED

For this reason, we say that f(α), f(α²), ...., and f(α^λ-1) are conjugates of each other.

4. Norm

Definition 2: Norm of a cyclotomic integer f(α)

Nf(α) = f(α)*f(α²)*...*f(α^λ-1)

I will now use this definition in the following proofs.

Lemma 3: Nf(α) = Nf(αⁱ) for all values of i between 1 and λ-1.

Proof:

(1) Nf(αⁱ) = f(αⁱ)*f(α^2*i)*...*f(α^i(λ-1))

(2) Now, each value i, 2*i, 3*i, ... (λ-1)*i maps to a distinct value of 1,2,3,...,(λ-1) modulo λ (see Lemma 1 here)

(3) So in each case, i,2*i, etc. maps to a₁*λ+1, a₂*λ+2, etc.

(4) So we get Nf(αⁱ) = f(α^a₀*λ+1)*f(α^a₁*λ+2)*...*f(α^{a_λ-1*λ+λ-1}) where a_i is a nonnegative integer.

(5) Since α^n*λ=1, we get:

Nf(αⁱ) = f(α)*f(α²)*...*f(α^λ-1)

QED

Lemma 4: α^j = α^λ-j

Proof:

(1) From roots of unity and Euler's Formula, we know that:

α = e^(i2π/λ) = cos(2π/λ) + isin(2π/λ)

(2) We also know that the complex conjugate of a + bi is a - bi, so the complex conjugate for α is:

α = cos(2π/λ) - isin(2π/λ)

(3) Likewise, we know that the complex conjugate for α^j is:

α^j = cos(2jπ/λ) - isin(2jπ/λ)

(4) Using Euler's Formula, we see that:

e^-2jπ/λ = cos(-2jπ/λ) + isin(-2jπ/λ)

(5) Since cos(-x) = cos(x) and sin(-x) = -sin(x) [see here], we can use (#4) to get:

e^-2jπ/λ = cos(2jπ/λ) - isin(2jπ/λ)

which is from #3, the complex conjugate for α^j

(6) Now, e^-2jπ/λ = (e^2π/λ)^-j =

= α^-j = α^-j*α^λ =

= α^{λ - j}

QED

Corollary 4.1: f(α^j) = f(α^λ-j)

Proof:

(1) From Lemma 1, we have:

f(α) = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

(2) From this,

f(α^j) = a₀ + a₁α^j + a₂α^2*j + ... a_λ-1α^j*(λ-1)

(3) Now, from Lemma 4, we know that:

f(α^j) = a₀ + a₁α^λ-j + a₂α^{λ - 2*j} + ... a_λ-1α^{λ - j*(λ-1)}

(4) And, we know that:

f(α^λ-j) = a₀ + a₁α^λ-j + a₂α^(λ-j)*2 + ... a_λ-1α^{(λ - j)*(λ - 1)}

(5) Now,

n*λ - j*n ≡ λ - j*n (mod λ) [See here if you need a review of modular arithmetic]

(6) So that we see that step #3 and step #4 are equal so that:

f(α^j) = f(α^λ-j)

QED

Corollary 4.2: f(α^j)*f(α^λ-j) is a nonnegative real number

Proof:

(1) f(α^j) * f(α^λ-j) = f(α^j)* f(α^j) [From Corollary 4.1 above]

(2) So that:
f(α^j) * f(α^λ-j) = (a₀ + a₁α^j + ... + a_λ-1α^j*(λ-1))(a₀ + a₁α^j + ... + a_λ-1α^j*(λ-1)) =

= (a₀)² + (a₁)²(α^j*α^j) + ... + (a_λ-1)²*α^j*(λ-1)*α^j*(λ-1))

(3) Since each α*α is a nonnegative number, the conclusion follows.

QED

Lemma 5: For any cyclotomic integer f(α), its norm is a nonnegative rational integer.

Proof:

(1) Using Lemma 1 above, we know that:

Nf(α) = a₀ + a₁α + a₂α² + ... + a_λ-1α^λ-1

(2) By Lemma 3 above, we can substitute any conjugate α^j and get the same norm so that:

Nf(α^j) = Nf(α)

(3) But by changing to a conjugate, we keep the same coefficients but get the following:

Nf(α^j) = a₀ + a₁α^j + a₂α^j*2 + ... + a_λ-1α^(λ-1)*j

(4) Combining the two equations gets us:

a₀ + a₁α^j + a₂α^j*2 + ... + a_λ-1α^(λ-1)*j = a₀ + a₁α + a₂α² + ... + a_λ-1αλ-1

(5) Subtracting one from the other gives us:

a₀ - a₀ + (a₁ - a_j)α^j + ... = 0

(6) Since we know that each of these j,2*j,...,(λ-1)*j matches up with a value 1,2,...,λ-1, we know that:

a₁ = a_j

(7) Further, since j can be any value from 2 thru λ-1, we can conclude the following:

a₁ = a₂ = a₃ = ... = a_λ-1

(8) So that:
Nf(α) = a₀ + a₁(α + α² + ... + α^λ-1)

(9) From Lemma 2, we know that:

1 + α+α² + ... + α^λ-1 = 0

so that:

α+α² + ... + α^λ-1 = -1

(10) So, we apply (#9) to (#8) to give us:

Nf(α) = a₀ - a₁

(11) We know that it is nonnegative since:

Nf(α) = [f(α¹)*f(α^λ-1)]*[f(α²)*f(α^λ-2)]*...

(12) From Corollary 4.2 above, we know that multiplication of (λ-1)/2 pairs of nonnegative values will result in a nonnegative value.

QED

Lemma 6: f(α)g(α) = h(α) → Nf(α)*Ng(α) = Nh(α)

Proof:

(1) Let f(α)g(α) = h(α)

(2) By Definition 2 above:

Nf(α) = f(α)*f(α²)*...*f(α^λ-1)

Ng(α) = g(α)*g(α²)*...*g(α^λ-1)

Nh(α) = h(α)*h(α²)*...*h(α^λ-1)

(3) Using step #1 gives us:

Nh(α) = f(α)*g(α)*f(α²)*g(α²)*...*f(α^λ-1)*g(α^λ-1) =

= f(α)*f(α²)*...*f(α^λ-1) *g(α)*g(α²)*...*g(α^λ-1) =

= Nf(α)*Ng(α)

QED

Fermat's Last Theorem: Proof for regular primes

One of the highpoints of the 19th century mathematics is Kummer's proof of Fermat's Last Theorem for regular primes.

Kummer's theory of ideal numbers is one of the foundations of algebraic number theory. In future blogs, I will talk about some of the other very important proofs that came out at this time (impossibility of a general method for quintic equations, transcendence of π, and the fundamental theorem of algebra) and show how Dedekind reinterpreted many of these developments into the modern concepts of ideals, rings, groups, and fields.

Kummer's proof comes down to three major points.

(A) For certain primes (which Kummer called "regular primes"), cyclotomic integers can be said to have a form of unique factorization. [See here for discussion on ideal numbers and how they "save" unique factorization for cyclotomic integers]

(B) For a regular prime λ, there is no solution to x^λ + y^λ = z^λ where x,y,z are pairwise relatively prime all prime to λ

(C) For a regular prime λ, there is no solution to x^λ + y^λ = z^λ where x,y, z are pairwise relatively prime and where λ divides z.

For the full proof, go here.

References

• Harold M. Edwards, Fermat's Last Theorem
  

Eudoxus of Cnidas

All original works of Eudoxus of Cnidas have been lost. All that we know is found in Euclid's Elements and referred to by commentators on Eudoxus. From this information, Eudoxus is now viewed as the second greatest mathematician of the ancient world after Archimedes.

I have spoken about the formal definition of the limit in a previous blog. Today's blog is part of the effort to review the background and details that make up, Euler's Identity, one of the most astounding mathematical formulas of all time:

e^iπ + 1 = 0

For those who are not familiar with Euler's Identity, you may wish to start here.

Eudoxus was born around 408 B.C. in Cnidas which is resides in modern Turkey. Not much is known about Eudoxus's early life. We do believe that he studied in Italy with Archytas who was one of the followers of Pythagorus. Archytas was fascinated by the problem of duplicating the cube (the problem of finding a compass/ruler construction for a cube root). It is also probable that Eudoxus studied Pythagorean number theory and music theory.

Later, Eudoxus studied medicine in Sicily under Philiston and later in Athens under Theomedan. While in Athens, he attended lectures by Plato and his followers at the newly created Academy.

Professor Heath writes that Eudoxus at this time was very poor (Heath quoted in MacTutor):

... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day.

Later, he went to Egypt to study astronomy with the priests at Heliopolis. He stayed in Egypt for over a year and after this time, he decided to open a school of his own in Cyzikus which is in Asia Minor near the shore the Sea of Marmara. The school became a big success.

There is evidence that after his school became established, Eudoxus went with some of his followers to visit Athens. At this time, there seems to have been a rivalry between Plato and Eudoxus. Eudoxus was critical of Plato's mathematical abilities and Plato seemed jealous of the popularity of Eudoxus's school.

Eudoxus later returned to Cnidas where he came one of the town leaders. He continued to write texts on theology, astronomy, and geography and built an observatory. Hipparchus refers to astronomic data captured by Eudoxus at this observatory that helped explain the rising and setting of constellations.

Eudoxus's contributions to mathematics are legendary. He proposed a theory of proportion to deal with irrational numbers which is found in Euclid's Elements, Book V. All rational numbers can be compared by finding a common unit. The problem is that with irrational numbers there is no such common unit. Eudoxus instead proposed the use of ranges so that a comparison occurs by applying a common multiple and then comparing the resultant value.

In modern terms, for real numbers a,b,c,d: a/b = c/d if for every pair of integers m, n:

(a) if ma is less than nb, then mc is less than nd.

(b) if ma = nb, then mc = nd

(c) if ma is greater than nb, then mc is greater than nd.

This work on proportions was very important in the modern definition of real numbers (see here).

In addition to his theory of proportion, he proposed what is today termed the Method of Exhaustion. This generalized the work done by Antiphon on estimating the area of a circle by using inscribed polygons. Eudoxus's theory is found in Euclid's Elements, Book X, Proposition 1. In Euclid, Eudoxus's Method of Exhaustion is used to show that the area of two circles is in proportion to the square of their diameters.

Eudoxus also offered a solution to the duplication of the cube using curved lines. Eratosthenes who wrote a history of the problem talks about Eudoxus's solution. Unfortunately, the exact details of his solution have been lost. Paul Tannery was able to come up with a proof that is consistent with information known about Eudoxus. The proof uses an algebraic curve known as a kampyle curve. Today, this solution is known as the kampyile of Eudoxus.

In his own lifetime, Eudoxus was most famous for his planetary theory where he was able to emulate the planetary motions using 27 spheres that turn upon each other. These spheres of Eudoxus are mentioned by Aristotle in his famous work Metaphysics.

Although his book Tour of the Earth has been lost, there still exist over 100 quotes in secondary sources. The book surveys all the different peoples known to Eudoxus and reviews their history, their political systems, and their culture. He wrote deeply about Egyptian society and about the Pythagoreans in Italy.

References

• Eudoxus, MacTutor Biography
• David Joyce, Euclid's Elements
  

Claudius Ptolemy

Claudius Ptolemy was born around 85AD in Egypt. He lived during a time of Hellenized Egypt but very little is known of his personal life.

His reputation rests on his unprecedented contributions to astronomy and geography and to the controversy that these works later caused as astronomers reacted to the famous work by Copernicus. In Ptolemy's universe, the earth sits at the center.

Ptolemy lived in Alexandria, Egypt. We know very little of his education. He mentions the astronomic data of Theon the mathematician who was probably Theon of Smyrna, one of his teachers. Many of his early works are dedicated to Syrus who may have also been one of his teachers.

Many of Ptolemy's major works still exist. His magnus opus is the Almagest which is Arabic for "the greatest". Its original name was the Mathematical Compilations. The Greeks soon started calling it The Greatest Compilations. Later on, the Arabic version became the basis of the Latin translation which is why Ptolemy's work is largely known as the Almagest.

The Almagest was one of Ptolemy's earliest works. It presents the Greek theory about the motions of the Sun, the moon, and the planets. This stood as the standard text on astronomy until Copernicus released his heliocentric theory in 1543.

It is a very ambitious work. Ptolemy writes (taken from MacTutor Biography):

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.

His geocentric astronomic model was based on Aristotle even though Aristarchus of Samos had argued for a heliocentric model. After presenting this model, he then introduces trigonometric models to predict the motions of the orbits. He presents tables of chords (which are similar to the sine function) and uses the concept of "epicycles" (circles-within-circles) to model the planetary movements. It is his theory of planets which is his greatest contribution. In his day, there were only 5 planets recognized and he presented detailed mathematical models for explaining their motions.

Although Hipparchus gets credit for much of the theory that Ptolemy presents, it is Ptolemy who was able to apply that theory to the planetary data.

The historian Toomer wrote (from MacTutor web site):

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

There have been many who questioned whether Ptolemy deserves his reputation for being the greatest astronomer of his day. Tycho Brahe noticed that all of Ptolemy's star catalogue data was consistently off by 1 degree longitude. Although Ptolemy claimed that he had gathered the data. Brahe believed that Ptolemy had copied the data from another source. Some scholars have argued that Hipparchus is the true genius and Ptolemy merely copied Hipparchus's work. It is very hard to assess the truth of any of these claims since almost of all of Hipparchus's work is lost.

Regardless of these criticisms, it is undeniable that Ptolemy played the lead role in the geocentric theory of the solar system that ruled the day until the publication of Copernicus's famous work. Hipparchus may be the father of trigonometry but it is Ptolemy who became the lead voice in the application of trigonometry to the motions of the planets, sun, and moon.

References

• Biography of Ptolemy, MacTutor
  

Hipparchus of Rhodes

Hipparchus is considered by many to be the father of trigonometry because he was the first to organize measurements in relation to angles in a trigonometric table. Unfortunately, all but one of his math writings have been lost and the one text that remains is a minor work. Most of what we know today comes from the Greek astronomer Ptolemy.

Hipparchus was born around 190 B.C. in the town of Nicaea and died around 120 B.C. in Rhodes. He may have studied geometry in Alexandria. Unfortunately, not much more is known about his life.

His influence on astronomy and mathematics has been significant. He produced the first trigonometric table based on chords (the table lists the length of the chord that corresponds to a given angle). He discovered the precession of the equinoxes and accurately determined the length of a year to within 6.5 minutes. It is believed that Hipparchus created the first star catalogue which may have consisted of 650 stars. It is believed that he created this around 134 B.C. when a new star was said to have burst across the sky.

He was able to calculate the distance of the moon from the earth and developed a theory of the moon's motions based on epicycles. He also used epicycles to model the Sun's motions. He was perhaps the first to be able to predict solar and lunar eclipses. Hipparchus may have invented the astrolabe and used an equatorial ring to observe the sun's equinoxes.

Most of what is known about Hipparchus is found in Ptolemy's magnus opus Almagest. It was not Ptolemy's intentions to preserve the memory of Hipparchus. In fact, Ptolemy seems to assume that his reader has access to Hipparchus's original works.

References

• Hipparchus, Wikipedia
• Hipparchus, MacTutor