| Inhoudsbeschrijving | Laatste nieuws | Studiemateriaal | Tentamen | College |
Docent:
Tanja Lange
e-mail:tanja@hyperelliptic.org
Instructeur:
Ruud Pellikaan (hoofd)
Leon Groot Bruinderink
Dit is the pagina voor het college 2WF50 - Algebra en Discrete Wiskunde. De oficiele pagina OWInfo.
Inhoudsbeschrijving
Monoiden en semigroepen, de classificatie van cyclische
monoiden. Productconstructies en deelstructuren. Groepen, de kleine
stelling van Fermat, de stelling van Lagrange over
ondergroepen. Ringen en lichamen, een uitgebreide kennismaking met
polynoomringen over een lichaam. Idealen en normale ondergroepen als
kernen van morfismen. De classificatie van eindige lichamen. Een meer
gedetailleerde studie van permutatie groepen, en automorfisme groepen
van combinatorische en meetkundige objecten.
Instructie: dinsdags 08:45 - 10:30, METAFORUM ZAAL 08
College: dinsdags 10:45 - 12:30, METAFORUM ZAAL 07, en vrijdags 13:45 - 15:30, LAPLACE-GEBOUW -1.19.
Op maart 10 geen instructie of college maar PRV62.
Algebra Interactive, Springer-verslag, Heidelberg 1999 (aanbevolen),
in engels. Dit boek is niet meer beschikbaar voor verkoop maar de
auteurs stellen een exemplaar voor download ter beschikking.
Meer informatie op het web
03 Feb 2015
Instructie:
Repetitie van rekenen modulo n, het
m-RSA-cryptosysteem, het Euclidische algoritme.
Public-key crypto
M-RSA
m-RSA is
"Kid-RSA" van Fellows en Koblitz gekraakt.
Attentie, did systeem is niet veilig.
College:
Monoiden en halfgroepen: definities, structuur,
n-tallige bewerkingen, halfgroepen, neutraal
element/eenheidselement.
Hier zijn de foto's van
het board.
07 Feb 2015
College:
Monoiden en halfgroepen: commutatieve bewerkingen,
deel-structuren (monoid, halfgroep), vermenigvuldigtafel, vrije
monoid over A, direct product van monoiden en halfgroepen, iedere
doorsnede van deelmonoiden/onderhalfgroepen is een monoid/halfgroep,
enkele voorbeelden.
Hier zijn de foto's van
het board.
10 Feb 2015
Instructie:
Exercises 6.7.3, 6.7.4, 6.7.5,6.7.7, 6.7.10 on pages 160, 161 of the
Algebra Interactive pdf file.
Huiswerk: Blad 1
College:
Homomorphismen, isomorphismen, example to show that f(e)=e'
is necessary,
cyclische monoiden, voortbrenger van een cyclische monoid,
<D>M: deelmonoid van M voortgebracht
door D, Ck,n
Hier zijn de foto's van
het board.
13 Feb 2013
College:
Inverse, Euler φ function,
φ(m)=|(Z/m)*|,
inverse van a
is uniek, schrapwet: uit xy=xz volgt y=z voor
inv(x) in M; groepen, ondergroepen.
Hier zijn de foto's van
het board.
24 Feb 2015
Instructie:
Exercises 6.7.11;
6.7.13;
6.7.16 with r= the real numbers;
6.7.24;
6.7.27 with R= the real numbers, GL(n,R) was denoted GLn(R) in
class and is the general linear group, i.e. the group of matrices with
nonzero determinant.
(pages 161 and 162 of the
Algebra Interactive pdf file).
Huiswerk: Blad 2
College:
Direct product van groepen,
cyclische groepen, voorbrenger, < g >, < D >,
Centrum Z(G), centralisator C(X,G),
normalisator N(X,G), morphismen, beeld, kern.
Voorbeelden:
27 Feb 2015
Doorsnede van ondergroepen is een ondergroep,
orde van een groep, orde van een element,
cyclische groepen,
voortbrenger, voorbeelden,
ondergroepen van cyclische groepen zijn cyclisch,
< g^k >=< g^d > voor d=gcd(k,n); < g^k > =G voor
gcd(k,n)=1.
.
Hier zijn de foto's van
het board.
03 Mar 2015
Block 1 en 2: Lecture given by Ruud Pellikaan.
Left cosets, Lagrange's Theorem, left cosets all have the same size,
order of an element divides the group oder, Fermat's little theorem,
normal subgroups, examples, kernel of a homomorphism is a normal
subgroup. Please read till the end of chapter 6 and check the
book for details that were skipped in the presentation.
Homework: Sheet 3. To hand in your
homework please email it as a .txt or .sage or as a sage worksheet
to Tanja. Do not email me a screenshot or printout, I will
try whether the program runs and takes my inputs.
Block 3 en 4: http://www.sagemath.org
introduction. Install sage or get an account before the exercise
round; bring your laptop.
Here is a quick reference
sheet for sage.
Unfortunately the connection to my sage server broke a few times
so some parts exist on the blackboard only.
Rick Sengers was so friendly to send me
photos of the blackboard.
Please take a look at the list of commands I
typed last year when I had the same
lecture. Note that some of them will give errors, these
are deliberate to show that only x is a predefined variable and that
Zmod.order() is not defined (could be addition or multiplication).
Today some of you pointed out that now order is defined as the additive
order; this might vary by version of sage.
06 March 2015
Lecture:
Definition of ring, lots of examples, including Gaussian integers,
n by n matrices over the reals, polynomial rings R[x]. Properties
of polynomials: degree, leading coefficient, monic, irreducible.
Subrings, properties of rings.
Here are the pictures
of the black board.
10 March 2015
No instructions and lectures but
Information Skills training. Make sure to bring your laptop.
13 March 2015
Examples of homomorphisms, Z[x]/(x^2+1) is isomorphic to Z[i], ideals,
kernel of a homomoprhism is an ideal, direct products, proof of
Chinese Remainder Theorem using direct products. (R×
S)×=R×× S×;
this can be used to prove Euler's phi function;
intersection of subrings is a subring, <D>R.
I didn't mention it in the lecture but the CRT also works for
polynomials instead of integers.
Here are the pictures
of the black board.
Homework: Sheet 4.
17 Mar 2015
Exercises:
Please prepare
6.7.31, 6.7.35,
7.8.1, 7.8.3, 7.8.4, 7.8.7, 7.8.8. You should use the exercise time
to ask questions about the parts that you find most complicated; no
need to go through these exercises in order.
Lecture:
Multiple, zero divisors, domains,
zero divisors are not invertible, if R is a domain then so is R[x],
cancellation law holds in domains, fields,
finite domains are fields,
L(a), field of fractions.
Read (and
understand) the proof that the quotient ring of a domain is a
field.
Here are the pictures
of the black board.
20 Mar 2015
Lecture:
Prime field, characterisitic, K is L-vectorspace, finite fields have
order equal to a prime power, field with 4 elements, a(x) invertible
in K[x]/(f(x)) if gcd(a,f)=1, if f is irreducible this gives a field,
α is a root of f(x) is equivalent to (x-α) divides f(x);
degree-n polynomial has at most n roots;
homomorphisms and fixed points, Frobenius automorphism (mapping of
x to x^q or x^p) for finite field and properties;
Notations for finite field as Fq.
Here are the pictures
of the black board.
Homework: Sheet 5.
24 Mar 2015
Exercises:
Please prepare 7.8.9, 7.8.11, 7.8.12, 7.8.13 (you can use the
homework), 7.8.14, 7.8.15. You should use the exercise time to ask
questions about the parts that you find most complicated; no need to
go through these exercises in order.
Homework: Sheet 6.
Updated the homework sheet to specify that you should please
send solutions to Leon.
Lecture:
Homomorphisms from field to a ring and properties,
algebraic integers; a algebraic over L; algebraic integers form a
field; ideals; intersection of ideals is an ideal, I+J is an ideal,
(V)R is an ideal, examples;
Z is a principal ideal domain = every
ideal is principal, i.e. generated by a single element; equivalent
statements to I=R; fields hve only trivial ideals,
prime ideals; maximal ideals; examples;
maximal ideals are prime ideals; over Z the converse almost holds,
only exception is (0)Z.
Here are the pictures
of the black board.
27 Mar 2015
Lecture:
residue classes; residue classes are
equivalence classes; R/I is a ring; first isomorphism theorem:
R/I is domain if I is a prime ideal;
R/I is field if I is a maximal ideal;
f homomorphism from ring R to ring S, then R/ker(f) is isomorphic to
Im(f), example: I=nZ or I=(x^2+x+1)F2
Finite fields:
for |L|=q have (x^q-x) is the product of (x-a), where a runs through all of L;
L* has q-1 elements; * notation is used for fields
to denote the multiplicative group;
subfields of field with p^n elements are such that they have
p^m elements, for m a divisor of n;
minimal polynomial of an algebraic element;
the minimal polynomial is unique and it is irreducible;
minimal polynomial of a has exactly all of a's conjugates as roots;
an irreducible polynomial over Fp of degree n
divides xpn-x; the latter equals the product of
all monic, irreducible polynomials of degree dividing n;
this gives a way to find irreducible polynomials and to prove that
for any positive integer n and any prime p there exists an irreducible
polynomial of degree n. I didn't get around to showing a nice example,
namely that there are (p^2-p)/2 irreducible polynomials of degree 2 over
Fp.
Quick repetition of how to solve CRT.
Please read in the book:
The multiplicative group is cyclic (note that the reference
should go to 6.5.10 and that the statement needs to hold for every
divisor of the group order, not just the group order itself);
the minimal polynomial of a divides every polynomial f which has
f(a)=0.
Here are the pictures
of the black board.
31 Mar 2015
Exercise (block 1+2, also block 3+4)
Please prepare:
7.8.19, 7.8.20, 7.8.21, 7.8.22, 7.8.23 (hint, use the norm),
7.8.24, 7.8.27 (what happens when you replace 3 by any prime? by any number?),
7.8.28, 7.8.30.
Het tentamen voor 2WF50 is gepland voor woensdag 15-04-2015
09:00-12:00.
De herkansing is gepland voor woensdag 01-07-2015, 18:00-21:00.
Tentamen: 70% van het eindresultaat
Huiswerkopdracht: 30% van het eindresultaat
Tentamen van 2015: 2015-04-15.pdf
Tentamen van 2014: 2014-04-16.pdf
en 2014-07-02.pdf
.
Tentamen van 2011: 2011-11-10.pdf en
2012-01-27.pdf.
Tentamen van 2010: 2010-11-02-exam.pdf en 2011-01-21-exam.pdf.
Voor de laatste jaren was Andries Brouwer de docent voor Algebra 2. Hij was zo vriendschappelijk om me zijn oude examens te geven.