Approaches to FLT
•
In the early approaches to FLT several
methods were used (See for instance the proof
of Germain Theorem).
1. Infinite descent
2. Congruences
3. Unique factorization into prime factors
•
In 1847 Lamé put forth a “proof” of Fermat
using a combination of the above techniques
which he thought to generalize.
A few weeks later Kummer showed that unique
factorization does not generalize in the way
that Lamé expected.
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Ernst Eduard Kummer
(1810-1893)
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• 1828 Kummer entered the University of Halle with the intention of
studying Protestant theology.
• 1831 Kummer was awarded a prize for a mathematical essay on a
topic set by Scherk. For this, he was also awarded a doctorate.
• 1831 became a teacher (for 10 years). Corresponded with Jacobi
and Dirichlet.
• 1836 elected into the Berlin Academy of Science.
• 1842 became a full professor in Breslau.
• 1855 became a professor in Berlin.
• 1883 retired because of “fading memory”.
• 1843 Kummer, realized that unique factorisation of integers did not
extend to other rings of complex numbers and hence attempts to
prove Fermat's Last Theorem broke down. To compensate he
introduced “ideal” numbers.
• 1857 Grand Prize of the Paris Academy of Sciences for his work
toward FLT, although he never entered the contest.
Complex numbers
• Complex numbers are pairs of real numbers (a,b) with the following
operations.
– (a,b)+(c,d)=(a+b,c+d)
– (a,b)(c,d)=(ac-bd,ad+bc)
• Notice that there is also subtraction and division.
• One can also formally introduce the symbol i with the relation i2=-1.
Then we can write complex numbers (a,b) as z=a+ib.
• Since a complex number is a pair of real numbers, it also has polar
coordinates (r,f).
– We can also write z=reif.
– To convert one uses eif=cos f + i sin f and r=|z|=(a2-b2)=(a+ib)(a-ib).
– Then if z=reif and w=seiy then zw=rsei(f+y).
• Fundamental Theorem of Algebra: Every polynomial equation of
degree n with complex coefficients has n roots in the complex
numbers.
• Example: zn-1=(z-1)(z-wn) (z-wn2) …(z-wnn-1)
–
wn=e2pi/n and wnn=e2pin/n=cos(2p)+isin(2p)=1.
Number systems
• Just like considering √-1, we can introduce √-5 and look
at the numbers a+b√-5, but now with a and b in Z.
• What changes? Unique factorization need not hold!
• 6=2·3=(1+√-5)(1- √-5)
• Also we need a new concept of prime numbers.
• We call m irreducible if m cannot be written as a product
not containing 1.
• We call p prime if p divides mn implies that either p
divides m or p divides n.
• Problem: In the above factorization all the factors are
irreducible, but none is a prime. So there is no way to
unique write 6 as a the product of primes.
Where Lamé went wrong
• Lamé factored xn+yn=(x+y)(x+wny) (x+wn2y) …(x+wnn-1y)
and then wanted to deduce that each linear factor has to
be an n-th power.
• Now this would be true if there were a unique
factorization.
• For n23 however Kummer showed that this is not
generally the case. For this consider numbers of the
form a0+a1w23+a2w232+…+a21w2321
(actually w2322=-1-w23-w232-…-w2321)
• But (1+w2+w4+w5+w10+w11)(1+w+w5+w6+w7+w9+w11) is
divisible by 2, but none of the factors are! So there is no
unique factorization!
Kummer’s “ideal numbers”
• Kummer thought of the failure of unique factorization as
caused by “hidden” divisors. He found a way to treat the
“hidden” divisors as “ideal” numbers.
• For this a number is generalized to the subset it
generates: 2  <2>={2n:n in Z}.
• Then we can also look at the sets of the form
A=<2,1+√-5>={2m+n(1+√-5):n,m in Z}
and think of this set as the common divisors of 2 and
1+√-5.
• We would get A·A =<4,2(1+√-5),-4+2√-5>=<2>.
• Setting B=<3, 1+√-5> and C=<3,1+√-5> one obtains
<6>=<2><3>=A2BC=ABAC=<1+√-5, 1-√-5>.