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The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of Algebraic Number Theory . THE REGULAR REPRESENTATION, TRACE AND DETERMINANT Suppose ''F'' is a Field Extension of the field of rational numbers Q of finite Degree ''n''. This means that ''F'' is an ''n''-dimensional Vector Space over Q, elements of ''F'' form a Commutative Ring under the operations of addition and multiplication, and all non-zero elements of ''F'' are invertible. Let us choose a Basis ''e''1, ..., ''e''''n'' for ''F'', then any element ''x'' of ''F'' has a unique representation in the form ''x'' = ∑ ''x''i ''e''i. Using the multiplication in ''F'', we may represent the elements of the field ''F'' by ''n'' by ''n'' matrices, as follows: : This way of associating a matrix to any element of the field ''F'' is called the regular representation. The Square Matrix ''A'' = ''A''(''x'') with the rational entries ''a''''ij'', where ''i'' and ''j'' are indices between 1 and ''n'', represents the effect of multiplication by ''x'' in the basis ''e''. It follows that if the element ''y'' of ''F'' is represented by a matrix ''B'', then the product ''xy'' is represented by the Matrix Product ''AB''. Invariant s of matrices, such as the Trace , Determinant , and Characteristic Polynomial , depend solely on the field element ''x'' and not on the basis. In particular, the trace of the matrix ''A''(''x'') is called the ''' Trace ''' of the field element ''x'' and denoted Tr(''x''), and the determinant is called the ''' Norm ''' of ''x'' and denoted N(''x''). Properties Let ''λ'' be a rational number, or as it is common to say, a ''scalar'', and ''x'', ''y'' be two elements of ''F'', then the trace and determinant have the following properties:
The first two properties express the fact that the trace is a linear function of ''x''. The third property is the multiplicativity of the norm, and the last property means that the norm is a Homogeneous Function of ''x'' of degree ''n''. ALGEBRAIC INTEGERS An element ''x'' of the algebraic number field ''F'' is called an Algebraic Integer if it is a root of a Monic Polynomial with integer coefficients. Algebraic integers admit other, equivalent descriptions. An element ''x'' of ''F'' is an algebraic integer if and only if the characteristic polynomial ''p''''A'' of the matrix ''A'' associated to ''x'' is a monic polynomial with integer coefficients. Suppose that the matrix ''A'' that represents an element ''x'' has integer entries in some basis ''e''. By the Cayley-Hamilton Theorem , ''p''''A''(''A'') = 0, and it follows that ''p''''A''(''x'') = 0, so that ''x'' is an algebraic integer. Conversely, if ''x'' is an element of ''F'' which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix ''A''. In this case it can be proven that ''A'' is an Integer Matrix in a suitable basis of ''F''. Note that the property of being an algebraic integer is ''defined'' in a way that is independent of a choice of a basis in ''F''. The set of integral square matrices is closed under addition and multiplication, and it follows that the algebraic integers in ''F'' form a Ring , denoted by ''O''''F'', which is a Subring of ''F''. A field contains no Zero Divisors and this property is inherited by any subring. Therefore, the ring of integers of ''F'' is an Integral Domain . The field ''F'' is the Field Of Fractions of the integral domain ''O''''F''. Properties
An abstract commutative ring with these three properties is called a Dedekind Ring (or '''Dedekind domain'''), in honor of Richard Dedekind , who undertook a deep study of rings of algebraic integers. BASES FOR NUMBER FIELDS Power basis Since there are only a finite number of subfields of ''F'', and since these correspond to subspaces of ''F'' as a vector space over Q, in general an element of ''F'' does not belong to any proper subfield, hence generates ''F'' and has an Irreducible minimal polynomial over Q. Such an element ''x'' is called a Primitive Element , and the Primitive Element Theorem tells us that extensions of fields of Characteristic zero indeed have a primitive element. If ''x'' is a primitive element, then ''x'', ''x''2, ..., ''x''''n'' − 1 is a basis for ''F''. If the characteristic polynomial for ''x'' has non-integral coefficients, then we may find the Greatest Common Divisor ''D'' of the denominators of the coefficients, and take instead the polynomial for ''y'' = ''Dx'' which we may obtain by substituting ''y''/''D'' for ''x'' in the polynomial for ''x''. This gives us an integral power basis, defined in terms of a single root of an irreducible Monic Polynomial of degree n over Q with integer coefficients. Integral basis An integral basis for a number field F of degree n is a set B = {b1,...,bn} of n algebraic integers in F such that every element of the ring of integers ''O''F of F can be written uniquely as a Z-linear combination of elements of B; that is, for any x in OF in we have x = m1b1+...+mnbn, where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as m1b1+...+mnbn, where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where the mi are all integers. Working Locally and using tools such as the Frobenius Map , it is always possible to explicitly compute such a basis, and it is now standard for Computer Algebra System s such as Maple and Mathematica to have built-in programs to do this. TRACE FORM AND DISCRIMINANT We may define a Bilinear Form on F by means of the trace, by T(''x'' ''y''); this is called the '''trace form'''. If ''b''1, ..., ''b''n is an integral basis for ''F'', then we may define a symmetric integral matrix, the '''integral trace form''', by ''t''ij = T(''b''i''b''j). Then the '''discriminant''' of ''F'' may be defined as det(''t''). It is an integer, and is an invariant property of the field ''F'', not depending on the choice of integral basis. Example Consider ''F'' = Q(''x''), where ''x'' satisfies ''x''3 − 11''x''2 + ''x'' + 1 = 0. Then an integral basis is ''x'', 1/2(''x''2 + 1) , and the corresponding integral trace form is : The determinant of this is 1304 = 23 163, the field discriminant; in comparison the Root Discriminant , or discriminant of the polynomial, is 5216 = 25 163. PLACES Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of P-adic Number s by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field F into its various topological completions at once. Archimedean places Given an irreducible polynomial ''f'' over Q defining a primitive element ''x'' of a number field ''F'', and hence a power basis for ''F'', we may factor ''f'' into irreducible factors over the real numbers '''R'''. These factors are either of degree one or two, and since there are no repeated roots, there are no repeated factors. Each factor of degree one gives a real root, and by replacing ''x'' by the real root ''r'', we obtain an embedding into the real numbers; the number of such embeddings is equal to the number of real roots. This allows us to define an Absolute Value on the elements of ''F'', since they are now elements of '''R'''; such an absolute value is called a '''real Place ''' of the number field ''F''. Similarly, for each factor of degree two we obtain a pair of conjugate complex numbers, which allows for two conjugate embeddings into '''C'''. Either one of this pair of embeddings can be used to define an absolute value on ''F'', which is the same for both embeddings since they are conjugate. This absolute value is called a '''complex place''' of ''F''. These are the Archimedean places of ''F'', corresponding to Archimedean Absolute Value s. Ultrametric places | ||
| Factoring The Polynomial ''f'' Of Degree ''n'' Satisfied By The Primitive Element ''x'', We Now May Obtain Factors Of Various Degrees, None Of Which Are Repeated, And The Degrees Of Which Add Up To ''n'' For Each Of These ''p''-adically Irreducible Factors ''t'', We May Suppose That ''x'' Satisfies ''t'' And Obtain An Embedding Of ''F'' Into An Algebraic Extension Of Finite Degree Over '''Q'''<sub>p</sub> Such A | "http://wwwinformationdelightinfo/information/entry/local_field" class="copylinks">Local Field behaves in many ways like a number field, and the ''p''-adic numbers may similarly play the role of the rationals in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to '''Q'''<sub>''p''</sub> By using this ''p''-adic norm map '''N'''<sub>''t''</sub> for the place ''t'', we may define an absolute value corresponding to a given ''p''-adically irreducible factor ''t'' of degree ''m'' by &theta<sub>''t''</sub> = '''N'''<sub>''t''</sub>(&theta)<sub>''p''</sub><sup>1/''m''</sup> Such an absolute value is called an Ultrametric , non-Archimedean or ''p''-adic place of ''F'' |
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| While This Corresponds To Less Than Three Digits Of Accuracy, The Factorization Is Easily Lifted To Much More Accurate Ones Involving Higher Powers Of 23, And In Any Case Already Suffices If We Consider The Element ''y'' | ''x'' &minus 10 of '''Q'''<sub>23</sub>, then by substituting ''x'' = ''y'' + 10 into the first factor ''f''<sub>1</sub> modulo 529, we obtain ''y'' + 191, so the valuation ''y''<sub>''f''<sub>1</sub> </sub> for ''y'' given by ''f''<sub>1</sub> is &minus191<sub>23</sub> = 1 On the other hand if we substitute ''x'' = ''y'' + 10 into ''f''<sub>2</sub>, we obtain ''y''<sup>2</sup> &minus 161''y'' &minus 161 modulo 529 Since 161 = 7×23, we find that |
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