Recommend Number Field with airSlate SignNow

Recommend number field

yeah this is a series of short talks what is phenomenal so this time we will have what is a number field by yourself [Music] Thanks so number fields that is my topic I started with q2 field off rice the numbers so if I have a field larger than this cue then that larger field the vector space or Q against consider a field cane which is contained in complex numbers and which contains faster numbers and this K is a vector space over Q in the natural mail now if the dimension of this field K is finite then we say that K is an algebraic number field so if k / q this dimension of K over Q as a vector space is finite we say that K is an algebraic number field though he said that this is called an algebraic number field is that its elements are algebraic numbers so any alpha in K is an algebraic number and by that we mean that this alpha satisfies a polynomial equation or Q so if we consider this dimension is n then for alpha and K the numbers 1 alpha alpha square up to alpha raise to n they are n plus 1 numbers and a dimensions and they are linearly dependent over the field Q that means they're in this a naught a 1 etc a n belonging to Q and not all 0 such that summation a I alpha raise to I I equal to 0 to n that is 0 so these are at least one of these is nonzero so I will just write some AI not equal to 0 examples of such number fields are Q I I is square root of -1 I can take Q root 7 I can take you tourists 1/3 here Q round bracket alpha this means all rational expressions in alpha with coefficients from Q so a polynomial in alpha upon a polynomial in alpha it is required that the loominator is a nonzero quantity and it can be seen that this Q around rocket alpha is in fact equal to a polynomial set of polynomials in alpha when alpha is an algebraic number this is an algebraic number field and interestingly any algebraic number field can be written in such a power so I will write down this you know that any day break number field can be written as few ALPA for some algebraic number no in this algebraic number field there are some special elements which are algebraic integers and these are algebraic numbers which satisfy a monic polynomial with integer coefficients so those which satisfy a unique polynomial with integer coefficients the set of all algebraic integers in an algebraic number field that is an integral domain and the examples of these de string of algebraic integers so examples are the following if I take Q this itself is an algebraic number field and the corresponding ring of algebraic integers is say any rational number which is also an algebraic integer has to be an integer usual integer if I take Q I the ring is red I these are all algebraic integers if I take Q root minus 3 then here this ring is not just red root minus 3 but if you want all algebraic integers you should write here 1 plus root minus 3/2 or you may also write it in this form whose head Omega where Omega is imaginary cube root of 1 this is minus 1 plus I root 3/2 and the ring of integers is an important object of study in algebraic number theory if you take Vita equal to Zeta n that is e raised to 2 pi I by n then this is this is two pi by n we have this complex number nearest to 2 pi by n and this is exactly nth part of this circumference of the circle and this circle is divided by this complex number we can say into n parts so this is the corresponding field k equal to Q Veta it is called as a cyclotomic field that it's circle cycle is circle and almost we spot so this car algebraic number or algebraic integer it divides the circle into n parts so here is 2 2 pi by n then 2 times 2 pi by n 3 times and so on you get n points in the circle which divide the circumference into n parts the ring of integers are equal to Z Zeta in this case it can be shown that the ring of integers is of this type and these rings of integers for various number fields satisfy very interesting properties these properties are the following one is that suppose R is equal to okay that is the ring of integers in the algebraic number field K then R is not area that is a we ideal in R is finitely generated second property is R is integrally closed so if any element in the quotient field of our that is K if it is integral over R then it has to be in R that is the property and the third property is every nonzero prime ideal of okay is maximum we know that these properties are there for any PID also so PID is a ring which satisfies all these properties anything which satisfies these properties is called as a Dedekind domain any domain which satisfies these three properties is called as a very keen domain and here an interesting property for ideals holds that is every nonzero non unit ideal that is not equal to the whole ring ideal in okay is a product of powers of prime areas moreover this happens uniquely so just as we have fundamental theorem of arithmetic every positive integer is a product of powers of primes here also we have this property but this property is therefore ideals clear our other nice properties of the ring of integers that is the ring of integers okay is a PID if only if it is you every this is one important property of okay you do a PID implant CFD but you already will not have implied PID here it is true that if you know that this is you every automatically it follows it is a PID another interesting fact about okay is every idea of okay can be generated by two elements can be helped by two elements so it is very close to being PA this is also a connection between Euclidean domain property and principle ideal domain property here furthering okay as we know every Euclidean domain is in principle ideal domain without the converse is true or not for okay that's a matter of investigation and if K is imaginary quadratic field field that is it is cue Rudy in that case these negative D the square free integer D less than zero and here we have so what are the pids that has been investigated for the ring of integers so okay is a PID if and only if V is equal to minus 1 minus 2 minus 3 minus 7 minus 11 minus 43 minus 19 - 43 - 67 minus 163 so these are line values for which it is a PID this was conducted by Gauss and later on it was proved by stark and Hitler so this is stark signal theorem out of which up to this we have Euclidean domains and these are not Euclidean domains they are only period so these are examples of PIDs which are not Euclidean device it is somewhat difficult to get such examples and after this imaginary quadratic fields one studies the remaining algebraic number fields for this property is okay a ukrainian domain provided you know it is a PID now [Music] there is a kind of conjecture so if generalized Riemann hypothesis pours generalized Riemann hypothesis for a specific set of holes and if the number of units in okay is infinite this just means that is K is not an imaginary quadratic field field then Weinberger has told that okay is the PID if and only if okay is a Euclidean domain so these societies and the generalized Riemann hypothesis and mathematicians are trying to prove this result without using the generalized Riemann hypothesis and there has been some success to this expectation thank you