Solvable Groups
... In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof that it is not always possible to express th ...
... In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof that it is not always possible to express th ...
The topological space of orderings of a rational function field
... Given a formally real field F, its set of orderings X(F) can be topologized to give a Boolean space (i.e., compact, Hausdorff and totally disconnected). This topological space arises naturally from looking at the Witt ring of the field, and has been studied by Knebusch, Rosenberg and Ware in the mor ...
... Given a formally real field F, its set of orderings X(F) can be topologized to give a Boolean space (i.e., compact, Hausdorff and totally disconnected). This topological space arises naturally from looking at the Witt ring of the field, and has been studied by Knebusch, Rosenberg and Ware in the mor ...
Quadratic reciprocity
... greater than or equal to 1, p1 , . . . pr distinct prime numbers and a1 , . . . ar are positive integers. The last (and non-trivial) result that we have to recall in order to proceed further is Fermat’s Little Theorem which says that if a is an integer and p is a prime, then ap ≡ a (mod p), or equiv ...
... greater than or equal to 1, p1 , . . . pr distinct prime numbers and a1 , . . . ar are positive integers. The last (and non-trivial) result that we have to recall in order to proceed further is Fermat’s Little Theorem which says that if a is an integer and p is a prime, then ap ≡ a (mod p), or equiv ...
Lecture Notes for Section 1.4 (Complex Numbers)
... For a complex number a + bi and its conjugate a – bi, theor product (a + bi)( a – bi), is the real number a2 + b2. To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, a ...
... For a complex number a + bi and its conjugate a – bi, theor product (a + bi)( a – bi), is the real number a2 + b2. To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, a ...
