Today's content is taken straight from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations which covers the history of Galois Theory from a mathematical perspective.

Definition 1: μ

_{p}

Let μ

_{p}denote the set of p-th roots of unity.

Examples:

μ

_{1}= {1}

μ

_{2}= {1, -1}

μ

_{3}= {1, (1/2)[-1 + √-3], 1/2)[-1 - √-3])

μ

_{4}= {1, -1, i, -i}

I will use μ

_{p}in the following context:

Definition 2: Q(μ

_{p})

Let Q(μ

_{p}) denote the set of complex numbers are rational expressions in these p-th roots of unity.

So that:

Definition 3: period of f terms of a p-th root of unity

For any two positive integers: e,f where ef = p-1, the periods of f terms are:

η

_{0}= ζ

_{0}+ ζ

_{e}+ ζ

_{2e}+ ... + ζ

_{e(f-1) }

η

_{1}= ζ

_{1}+ ζ

_{e+1}+ ζ

_{2e+1}+ ... + ζ

_{e(f-1)+1}

η

_{2}= ζ

_{2}+ ζ

_{e+2}+ ζ

_{2e+2}+ ... + ζ

_{e(f-1)+2}

...

η

_{e-1}= ζ

_{e-1}+ ζ

_{2e-1}+ ζ

_{3e-1}+ ... + ζ

_{e(f-1)+(e-1)}

If you review Alexander-Theophile Vandermonde's solution of the eleventh root of unity, it is clear that Carl Friedrich Gauss's theory of periods is a generalization of his solution. Interestingly, it is not clear if Gauss derived his solution from the work of Vandermonde or if he came upon it independently as part of his solution of the seventeenth root of unity.

Definition 4: σ

^{i}(f)

If we number each time that we apply σ such that σ(σ(f)) = σ

_{1}(σ

_{2}(f)), then:

σ

^{i}(f) = σ

_{1}(σ

_{2}(...(σ

_{i}(f))...))

We will now use the equation ef=p-1 (see Definition 3 above) to define a set whose elements are invariant under σ

^{e}.

Definition 5: K

_{f}

By K

_{f}, let us denote the set of all Q(μ

_{p}) which are invariant under σ

^{e}where ef = p-1.

Examples of K

_{f}:

(1) All rational numbers

u ∈ Q → u ∈ K

_{f}[This is clear from Lemma 6, here]

(2) All periods of cyclotomic equations

This is clear from definition 3 above.

Now, we can use these definitions to identify some properties which we will use later.

Theorem 1: K

_{f}is a vector space

Proof:

The proof follows from Definition 2, here since:

(1) K

_{f}is nonempty [See Lemma 6, here since Q is nonempty]

(2) K

_{f}is closed on addition. [See Lemma 5, here]

(3) K

_{f}is closed on scalar multiplication. [See Lemma 7, here]

(4) K

_{f}addition is associative. [See Lemma 5, here]

(5) 0 ∈ K

_{f}[See Lemma 6, here since 0 ∈ Q]

(6) K

_{f}has negative elements [See Lemma 7, here since (-1)*σ(x) = σ(-x)]

(7) K

_{f}addition is commutative. [See Lemma 5, here]

(8) All elements of K

_{f}are distributive since:

σ(a[b + c]) = σ(ab + ac)

(9) Scalar multiplication is associative [See Lemma 7, here]

(10) Existence of 1 [See Lemma 6, here since 1 ∈ Q]

QED

Theorem 2: Every element in K

_{f}can be written in a unique way as a linear combination with rational coefficients of the e periods of f terms.

(1) Let a be an arbitrary element in K

_{f}[See Definition 5 above]

(2) We can write a as follows: [See Definition 2 above and See Definition 1, here, for ζ

_{i}]

a = a

_{0}ζ

_{0}+ a

_{1}ζ

_{1}+ ... + a

_{e-1}ζ

_{e-1}+

+ a

_{e}ζ

_{e}+ a

_{e+1}ζ

_{e+1}+ ... + a

_{2e-1}ζ

_{2e-1}+

+ ... +

+ a

_{e(f-1)}ζ

_{e(f-1)}+ a

_{e(f-1)+1}ζ

_{e(f-1)+1}+ ... + a

_{p-2}ζ

_{p-2}

(3) By the definition of σ

^{i}[See Definition 4 above], we have:

σ

^{e}(a) = a

_{0}ζ

_{e}+ a

_{1}ζ

_{e+1}+ ... + a

_{e-1}ζ

_{2e-1}+

+ a

_{eζ}

_{2e}+ a

_{e+1}ζ

_{2e+1}+ ... + a

_{2e-1}ζ

_{3e-1}+

+ ... +

+ a

_{e(f-1)}ζ

_{0}+ a

_{e(f-1)+1}ζ

_{1}+ ... + a

_{p-2}ζ

_{e-1}.

(4) Since a ∈ K

_{f}, we know that:

σ

^{e}(a) = a

(5) Thus:

a

_{0}= a

_{e}= a

_{2e}= ... = a

_{e(f-1)}

a

_{1}= a

_{e+1}= a

_{2e+1}= ... = a

_{e(f-1)+1}

...

a

_{e-1}= a

_{2e-1}= a

_{3e-1}= ... = a

_{p-2}

(6) Therefore:

a = a

_{0}(ζ

_{0}+ ζ

_{e}+ ... + ζ

_{e(f-1)}) +

+ a

_{1}(ζ

_{1}+ ζ

_{e+1}+ ... + ζ

_{e(f-1)+1}) +

+ ... +

+ a

_{e-1}(ζ

_{e-1}+ ζ

_{2e-1}+ ... + ζ

_{p-2}).

(7) This proves that a is a linear combination of the periods, since the expressions between the brackets are the periods of f terms. [See Definition 3 above]

(8) Further, this expression is unique. [See Theorem 4, here]

QED

Corollary 2.1: 1, η, η

_{2}, ..., η

_{e-1}is a basis for K

_{f}

Proof:

This follows directly from Theorem 1 above, Theorem 2 above and Lemma 1, here.

QED

Theorem 3:

1, η, η

^{2}, ..., η

^{e-1 }is a basis for the vector space K

_{f}

Proof:

(1) 1, η, η

^{2}, ..., η

^{e-1}are linearly independent [see Definition 1, here for definition of linearly independent if needed] since:

(a) Assume that a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}= 0 for some rational numbers a

_{0}, ..., a

_{e-1}

(b) Then η is the root of the polynomial p(x) where:

p(x) = a

_{0}+ a

_{1}x + ... + a

_{e-1}x

^{e-1}(from step #1a)

(c) Now if a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}= 0, it follows that:

σ(a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}) = σ(0) = 0

σ

^{2}(a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}) = σ

^{2}(0) = 0

σ

^{3}(a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}) = σ

^{3}(0) = 0

...

σ

^{e-1}(a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}) = σ

^{e-1}(0) = 0

(d) So, σ(η), σ

^{2}(η), ..., σ

^{e-1}(η) are all roots of p(x) in step #1b

(e) Now, each of η, σ(η), etc. are the e periods of f terms which are pairwise distinct [See Definition 3 above]

(f) Since by the Fundamental Theorem of Algebra (see Theorem, here), the polynomial p(x) has degree at most e -1, it cannot have as roots the e periods of the f terms unless it is the zero polynomial.

(g) Therefore, a

_{0}= ... = a

_{e-1}= 0

(h) This then proves 1, η, η

^{2}, ..., η

^{e-1}are linearly independent. [See Definition 1, here]

(2) From Corollary 2.1 above, we know that dim K

_{f}= e. [See Theorem 1, here and Definition 2, here]

(3) But then using the fact 1, η, η

^{2}, ..., η

^{e-1}are linearly independent and Lemma 2, here, we can conclude that:

1, η, η

^{2}, ..., η

^{e-1}is a basis for K

_{f}.

QED

Corollary 3.1:

If η, η' are periods of f terms, then:

η' = a

_{0}+ a

_{1}η + ... + a

_{e-1}η

^{e-1}

for some rational numbers a

_{0}, ..., a

_{e-1}

Proof:

This follows from Theorem 3 above since η' ∈ K

_{f}and 1, η, η

^{2}, ..., η

^{e-1 }is a basis for the vector space K

_{f}.

QED

Lemma 4:

if gh=ef=p-1 and f divides g, then it follows that:

K

_{g}⊂ K

_{f}

Proof:

(1) Since gh=ef and f divides g, there exists an integer k such that:

k = g/f = e/h

(2) Therefore e = hk which gives us that:

σ

^{e}= (σ

^{h})

^{k}

(3) This means that every element that is invariant under σ

^{h}is also invariant under σ

^{e}since:

(a) Assume that an element a is invariant under σ

^{h}such that:

σ

^{h}(a) = a

(b) Further:

σ

^{h1}(σ

^{h2}(...(σ

^{hk}(a)...))) = a

(4) Since h*k = e, it follows from definition 4 above that:

σ

^{h1}(σ

^{h2}(...(σ

^{hk}(a)...))) = σ

^{e}(a)

(5) And it follows that:

σ

^{e}(a) = a

(6) Since σ

^{h}(a) = a → a ∈ K

_{g}and σ

^{e}(a) = a → a ∈ K

_{f}, it follows that:

K

_{g}⊂ K

_{f}

QED

Lemma 5:

Let f,g be divisors of p-1.

If f divides g, then every element in K

_{f}is a root of a polynomial of degree g/f with coefficients in K

_{g}

Proof:

(1) Let a be an element of K

_{f}

(2) Let us define k such that:

k = g/f

Since ef = gh, it follows that:

k = g/f = e/h

(3) Let use define P(x) such that:

P(x) = (x - a)(x - σ

^{h}(a))(x - σ

^{2h}(a))*...*(x - σ

^{h(k-1)}(a))

(4) P(x) has degree hk/h = k = g/f

(5) It is also clear that a is a root of P(x). [Since if x=a, then P(x)=0]

(6) We note that:

σ

^{h}(σ

^{h(k-1)}(a)) = σ

^{hk}(a) = (σ

^{h}(a))

^{k}

(7) Since k = e/h, it follows that e=hk and:

(σ

^{h}(a))

^{k }= σ

^{e}(a) = a

(8) Step #3 and step #6 and step #7 give us that:

σ

^{h}(P(x)) = P(x)

(9) Therefore, we conclude that P(x) has coefficients in K

_{g}.

QED

Corollary 5.1:

Let f,g be divisors of p-1 and let η, ξ be periods of f and g terms respectively.

If f divides g, then η is a root of a polynomial of degree g/f whose coefficients are rational expressions of ξ

Proof:

(1) ξ ∈ K

_{g}, and η ∈ K

_{f}

(2) Using Lemma 5 above, we know that η is a root of a polynomial P(x) of degree g/f with coefficients in K

_{g}

(3) Using Theorem 3 above, it follows that:

P(x) has coefficients which are rational expressions of ξ.

QED

References

- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001