By Edward Burger
2 DVD set with 24 lectures half-hour every one for a complete of 720 minutes...Performers: Taught through: Professor Edward B. Burger, Williams College.Annotation Lectures 1-12 of 24."Course No. 1495"Lecture 1. quantity concept and mathematical study -- lecture 2. usual numbers and their personalities -- lecture three. Triangular numbers and their progressions -- lecture four. Geometric progressions, exponential development -- lecture five. Recurrence sequences -- lecture 6. The Binet formulation and towers of Hanoi -- lecture 7. The classical concept of top numbers -- lecture eight. Euler's product formulation and divisibility -- lecture nine. The major quantity theorem and Riemann -- lecture 10. department set of rules and modular mathematics -- lecture eleven. Cryptography and Fermat's little theorem -- lecture 12. The RSA encryption scheme.Summary Professor Burger starts off with an summary of the high-level strategies. subsequent, he offers a step by step rationalization of the formulation and calculations that lay on the middle of every conundrum. via transparent factors, pleasing anecdotes, and enlightening demonstrations, Professor Burger makes this exciting box of analysis available for somebody who appreciates the attention-grabbing nature of numbers. -- writer.
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Extra resources for An Introduction to Number Theory (Guidebook, parts 1,2)
Sekiguchi) a) Let K be as above and for m > 2 , let 01 B B B fB 2 Mm (K ) : B = B B B @ b1 .. Bm (K ) denote the set b2 bm;1 1 .. . 0 . . b2 b1 C C C C = [1 b1 C C A : : : bm;1 ]g 1 b) 7. Show that Bm (K ) is a subgroup of GLm (K ) . Let h: Bm (K ) ! 1 + XK [[X ]]=(1 + X m K [[X ]]) be defined as h([1 b1 : : : bm;1 ]) = 1 + b1 X + + bm;1 X m;1 mod 1 + X m K [[X ]]: Show that h is an isomorphism. c) Let gm : 1+ XK [[X ]] ! Bm (K ) be the surjective homomorphism induced by h and let fn : W (K ) !
I show that there exists an element = i>0 ai X i 2 F p ((X )) which is not algebraic over Fp (X ) . b) Let = p and let F be the separable algebraic closure of Fp (X )( ) in Fp ((X )) . Show that F is dense in Fp ((X )) and Henselian. Let L = F ( ) . Show that L=F is of degree p , and that the index of ramification and the residue degree of L=F are equal to 1. Let F be a field with a discrete valuation v . Show that the following conditions are equivalent: (1) F is a Henselian discrete valuation field.
Then we get (;1)m a0 = NF ( )=F ( ) and if s = jL : F ( )j , then ((;1)m a0 )s = NL=F ( ) . 4), we get v (ai ) > 0 for 0 6 i 6 m ; 1 . However, (;1)m NF ( )=F (1 + ) = f (;1) = (;1)m + am;1 (;1)m;1 + + a0 and it suffices to show that if hence ; v NF ( )=F (1 + ) >0 and ; v NL=F (1 + ) >0 II. , w0 (1 + ) > 0 . Thus, we have shown that w0 is a valuation on L . Let n = jL : F j ; then w0 ( ) = nv ( ) for 2 F . Hence, the valuation (1=n)w0 is an extension of v to L (note that (1=n)w0 (L ) 6= Z in general).
An Introduction to Number Theory (Guidebook, parts 1,2) by Edward Burger