A short presentation of the course Algebraic number theory and Galois theory

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A short presentation of the course Algebraic number theory and Galois theory

Sara Checcoli (Institut Fourier, Grenoble)

M2R-Maths-Fonda 2017-2018

’Number theory and Geometry’

UGA

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Topics

é Part I: Galois theory;

é Part II: Introduction top-adic fields.

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Topics

é Part I: Galois theory;

é Part II: Introduction top-adic fields.

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Part I: Galois theory - Introduction

Letf(x)2 Q[x]be an irreducible polynomial.

Question: Are there "simple" general formulae to express the roots off(x)?

é degf(x) =2: Yes(formula by Brahmagupta, 628 ca.) f(x) =ax²+bx+c !x₁_;₂=^x^b^~

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é degf(x) =3: Yes(formulae by Tartaglia, 1539)

é degf(x) =4: Yes(formulae by Ferrari and Cardano, 1540) All these formulas only uses+; x; z; |;pⁿ

!if degf(x)4,f(x) =0 can besolved by radicals.

Question: Is this always the case for higher degree polynomials? No!

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Part I: Galois theory - Introduction

pb²x4ac 2a

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Part I: Galois theory - Introduction

é degf(x) =2: Yes(formula by Brahmagupta, 628 ca.)

f(x) =ax²+bx+c !x₁_;₂=^x^b^~

pb²x4ac 2a

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Part I: Galois theory - Introduction

é degf(x) =2: Yes(formula by Brahmagupta, 628 ca.) f(x) =ax²+bx+c

!x₁_;₂=^x^b^~

pb²x4ac 2a

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Part I: Galois theory - Introduction

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Part I: Galois theory - Introduction

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Part I: Galois theory - Introduction

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é degf(x) =4: Yes(formulae by Ferrari and Cardano, 1540)

All these formulas only uses+; x; z; |;pⁿ

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Part I: Galois theory - Introduction

pb²x4ac 2a

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Part I: Galois theory - Introduction

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Part I: Galois theory - Introduction

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Question: Is this always the case for higher degree polynomials?

No!

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Part I: Galois theory - Introduction

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Question: Is this always the case for higher degree polynomials?

No!

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Part I: Galois theory - Introduction

Theorem (Abel-Ruffini)There is no general formula in radicals for the roots that works for all polynomials of a given degree5.

Questions: What’s behind this phenomenon? Evariste Galois

(Bourg la Reine, 1811 - Paris, 1832)

Studied the "symmetries" among the roots & ’invented’ group theory

é {permutations of the roots off(x) fixing the algebraic relations with coefficients in Q among the roots} is a group, the Galois group of f(x).

é f(x) =0 is solvable by radicals, the Galois group off(x) is solvable.

é The Galois group of a generic polynomial of degree n is S_n, not solvable for n5 !Abel-Ruffini.

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Part I: Galois theory - Introduction

Questions: What’s behind this phenomenon?

Evariste Galois

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Part I: Galois theory - Introduction

Evariste Galois

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Part I: Galois theory - Introduction

Evariste Galois

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Part I: Galois theory - Introduction

Evariste Galois

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Part I: Galois theory - Introduction

Evariste Galois

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Part I: Galois theory - Introduction

Evariste Galois

é The Galois group of a generic polynomial of degree n is S_n, not solvable for n5

!Abel-Ruffini.

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Part I: Galois theory - Introduction

Evariste Galois

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Part I: Galois theory - Course content

Highlights:

é Introduce Galois extensions (over general fields) and their Galois groups

é Prove the fundamental theorem of Galois theory (Galois correspondence)

é Prove the generalisation of Abel-Ruffini theorem about extensions solvable by radicals

é Inverse Galois problem: given a finite group G, is there an irreducible polynomial in Q[x]having G as Galois group? Open! !Some cases where the answer is yes.

é (If time) Some elements of Galois theory for infinite extensions

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Part I: Galois theory - Course content

Highlights:

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Part I: Galois theory - Course content

Highlights:

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Part I: Galois theory - Course content

Highlights:

é Inverse Galois problem: given a finite group G, is there an irreducible polynomial in Q[x]having G as Galois group?

Open! !Some cases where the answer is yes.

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Part I: Galois theory - Course content

Highlights:

Open!

!Some cases where the answer is yes.

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Part I: Galois theory - Course content

Highlights:

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Part I: Galois theory - Course content

Highlights:

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Part II: p-adic fields - Introduction

Distances onQ:

Fix a prime numberp.

é Usual distance: x andy inQ are close if jxxyjis small.

Completion ofQ with respect to j y j

R

é p-adic distance: x andy in Q are close if xxy is highly divisible byp

(i.e. xxy=p^v(a=b) where (p;ab) =1 andv is big).

Completion ofQ with respect to j y jp

Qp the field of p-adic numbers

Origins:

é Hensel (1897): bring the methods of power series into number theory

(mimic the Taylor series expansion of rational functions in C(x) for rational numbers)

é Qp=^P_n_n₀a_npⁿj0a_ipx1;n₀2 Z

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Part II: p-adic fields - Introduction

Distances onQ: Fix a prime numberp.

R

é p-adic distance: x andy in Q are close if xxy is highly divisible byp

Origins:

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Part II: p-adic fields - Introduction

R

é p-adic distance: x andy in Q are close ifxxy is highly divisible byp

Origins:

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Part II: p-adic fields - Introduction

R

é p-adic distance: x andy in Q are close ifxxy is highly divisible byp

Origins:

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Part II: p-adic fields - Introduction

R

é p-adic distance: x andy in Q are close ifxxy is highly divisible byp (i.e. xxy=p^v(a=b) where (p;ab) =1 andv is big).

Origins:

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Part II: p-adic fields - Introduction

R

Origins:

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Part II: p-adic fields - Introduction

Completion ofQ with respect to j y j R

Origins:

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Part II: p-adic fields - Introduction

Origins:

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Part II: p-adic fields - Introduction

Completion ofQ with respect to j y jp Qp the field of p-adic numbers

Origins:

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Part II: p-adic fields - Introduction

Origins:

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Part II: p-adic fields - Introduction

Origins:

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Part II: p-adic fields - Introduction

Origins:

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Part II: p-adic fields - Introduction

Origins:

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Part II: p-adic fields - Introduction

p-adic fields: Finite extensions of Qp.

é K=Qnumber field

é prime ideal of K above p

é j y j-adic absolute value

Completion of(K; j y j)K=Qp a p-adic field. What arep-adic fields good for? Just few examples

é Hasse local-global principle: Let Q(x1; : : : ;x_n)be a

quadratic form with coefficients in Q. Then Q(x₁; : : : ;x_n) =0 has a non trivial solution in Q , it has a solution inRand in Qp for all p.

!applications to diophantine equations

é Monsky’s theorem: It is not possible to dissect a square into an odd number of triangles of equal area (the proof uses Q2)

é Galois representations

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é j y j-adic absolute value Completion of(K; j y j)

K=Qp a p-adic field. What arep-adic fields good for? Just few examples

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

Completion of(K; j y j)K=Qp a p-adic field.

What arep-adic fields good for? Just few examples

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

What arep-adic fields good for?

Just few examples

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Hasse local-global principle: Let Q(x1; : : : ;x_n)be a quadratic form with coefficients in Q.

Then Q(x₁; : : : ;x_n) =0 has a non trivial solution in Q , it has a solution inRand in Qp for all p.

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

quadratic form with coefficients in Q. ThenQ(x₁; : : : ;x_n) =0 has a non trivial solution in Q , it has a solution inR and in Qp for all p.

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Monsky’s theorem: It is not possible to dissect a square into an odd number of triangles of equal area

(the proof usesQ2)

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Introduction

é K=Qnumber field

é Berkovich spaces

é ...

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Part II: p-adic fields - Content of the course

Highlights:

é Recall of basic results on number fields (ring of integers, splitting of primes in extensions,. . . ) [depending on the audience]

é Definition ofp-adic fields and their properties (in particular classification of unramified, tamely ramified and wildly ramified extensions)

é Galois extensions ofp-adic fields: study the structure of their Galois group

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Part II: p-adic fields - Content of the course

Highlights:

é Definition ofp-adic fields and their properties

(in particular classification of unramified, tamely ramified and wildly ramified extensions)

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Part II: p-adic fields - Content of the course

Highlights:

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Part II: p-adic fields - Content of the course

Highlights:

é Galois extensions ofp-adic fields:

study the structure of their Galois group

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Part II: p-adic fields - Content of the course

Highlights: