A Brief History of Algebra

The Greeks: Euclid, Pythagora, Archimedes •

Indian and arab mathematicians •

Italian mathematics in the Renaissance •

The Fundamental Theorem of Algebra •

Hilbert’s problems •

1 Pythagoras, 500 BC

Pythagorean triplets: known to babylonians, mayans and hindus.

a2 + b2 = c2

Examples: (120, 119, 169), (3456, 3367, 4825), (13500, 12709, 18541). These appear in the Plimpton 322 tablet, Hammurapi dinasty. Clearly they must have had the algorithm to construct them. The triplets appear on manuals for con- struction of temples and sacred buildings.

Irrational numbers: the proof √2 is not ra- tional.

2 Euclid, 300 BC

The 13 books of the elements describe geo- metrical facts about triangle circles and other planar and spatial figures.

It is the first axiomatic exposition of mathe- matics (nowadays the only approach used).

Parallel postulate: Given a line ℓ and a point P not on ℓ, then there exists exactly one line through P that does not meet ℓ.

Euclid felt unconfortable in using this axiom and tried to prove as much as possible without using it.

Non-euclidean geometry: Gauss, Bolyai, Lo- batchevski.

3 Archimedes 287-212 BC Diophantus 250 BC

Archimedes was the first to use the symbol π and gave the approximation: 10 1 3+ <π< 3+ 71 7

Archimedes invented the fluxion method to compute the volume of certain solids. This is closely related to integration as we know it and contains the seminal idea of limit.

Diophantus was the first to examine the in- tegral and rational solutions of equations of type: xn + yn = zn, Last Fermat’s Theorem x2 Ny2 = 1, Pell’s equation − ±

4 Indian and Arab mathematics, V I XII − century AD

Bramagupta (VI century) made astonishing as- tronomical predictions on the distant stars.

Bhaskara (1114-1813 AD) discovered the Cakravala, and algorithm to solve the Pell’s equation (the complete treatment was done by Lagrange 1736-1813).

The arabs carried through the centuries the books by Euclid, which were never translated into latin!

Al-Kwaritzmi (780-850 AD) wrote the formula for the solution of the second degree equa- tion (known much before his time) in his book Al-jabr. Any formula is proved algebraically and/or geometrically.

5 Beginnings Algebra in Europe

Fibonacci (Filio Bonacci, 1180-1240) published three important works: 1. Liber Abbaci (1202) 2. Flos (1225) 3. Liber Quadratorum

Types of questions addressed: Find a number X such that X2+5 and X2 5 are both squares. −

The Fibonacci Numbers: 1, 1, 2, 3, 5, 8,...

XN = XN 1 + XN 2, X0 = X1 = 1 − − Formula:

X = (1+ √5/2)N + (1 √5/2)N N −

Problem: How many sheeps can we have after 5 years if we start with one sheep and each sheep produces one every year and from the second year on is productive? 6 Fra Luca Pacioli (1445-1514) wrote Summa de aritmetica, geometrica proportioni e propor- tionabilita’ in italian. He started using notation for expressions like:

q40 √320 RV 40mR320 − R meant “radical”, RV meant “apply radical to the whole expression” and so on.

7 The Bologna’s School: cubic and biquadratic equations

Scipione Del Ferro (Bologna 1465-1526) solved the equation: X3 + pX = q but did not publish his results!

Antonio Maria Fior (Del Ferro’s pupil) and Nic- colo’ Tartaglia issued public challenges to each other: the loser had to pay dinner for 30 peo- ple.

Gerolamo Cardano (Milan 1501-1576) published the book “Ars Magna” with the solution of third and second degree polynomial equations. These formulas are now known as Cardano’s formulas.

Cardano gave the first definition of imaginary number. The complex numbers were later ex- tensively studied by Gauss (1777-1885).

8 The Fundamental Theorem of Algebra

Given any polynomial P with complex coeffi- cients,

N N 1 N 2 P (z)= z + a1z − + a2z − + ...aN 1z + aN − there are unique (up to a permutation) com- plex numbers r1 ...rN , not necessarily distinct, such that:

P (z) = (z r )(z r ) ... (z r ). − 1 − 2 − N

The methods used by Gauss to prove this theo- rem form the foundation of the theory of func- tions of a complex variable. No algebraic proof is available!

9 Real version of the fundamental theorem of algebra:

Given any polynomial P with real coefficients,

N N 1 N 2 P (z)= z + a1z − + a2z − + ...aN 1z + aN − then:

P (z) = (z r1)(z r2) ... (z rk) 2 − − 2 − (z + 2s1z + t1) ... (z + 2smz + tm) where ri, sj, tl are real numbers and either k or m can be zero.

No formulas are available for degree greater than 4. This remarkable discover is due to Galois (1811-1832) who founded Galois theory (at age 17!), Abel (1802-1829), Ruffini (1765- 1822).

These are the so called no go theorems.

10 Hilbert problems

The birth of modern algebra (around 1900) is regarded to take place with Hilbert basis the- orem:

If A = C[x1 ...xn] is the algebra of polynomials in the variables x1 ...xn with coefficients in C, then every ideal I A is finitely generated. ⊂

During the Hilbert’s presentation of the proof of this result Gordan exclaimed: This is not mathematics, it is theology!

Hilbert presented in 1900 a list of 23 prob- lems for mathematicians to work on. The second problem is to give to mathematics a sound foundation so that every mathematical statement can be resolved to be either true or false. To everybody great surprise Goedel (1906-1978) proved that in mathematics there

11 are statements that are true but cannot be demonstrated to be so. This is the famous Goedel incompleteness Theorem.

Some of the Hilbert problems are keeping math- ematicians busy today like the 8th problem also known as the Riemann hypothesis:

The real part of any non-trivial zero of the Riemann zeta function is .