In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. An example is the mapping from the complex numbers to the real numbers sending Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Field theory is a branch of mathematics which studies the properties of fields. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Please refer to Real vs. ...

x + iy

to

x² + y².

In general if K is a field and L a Galois extension of K, the norm N_L/K of an element α of L is defined as the product of all the conjugates In mathematics, a Galois extension is a field extension that has a Galois group. ... In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial PK,α(t) of α over K. If K is given inside an algebraically closed field C, then the conjugates can be taken inside...

g(α)

of α, for g in the Galois group G of L/K. Since In mathematics, a Galois group is a group associated with a certain type of field extension. ...

N_L/K(α)

is immediately seen to be invariant under G, it follows that it lies in K. It also follows directly from the definition that

N_L/K(αβ) = N_L/K(α)N_L/K(β)

so that the norm, when considered on non-zero elements, is a group homomorphism from the multiplicative group of L to that of K. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

The norm of an algebraic element γ over K can be defined directly as the product N(γ) of the roots of its minimal polynomial. Assuming γ is in L, the elements The minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0. ...

g(γ)

are those roots, each repeated a certain number d of times. Here

d = [L: M]

is the degree of L over the subfield M of L that is the splitting field of the minimal polynomial of γ. Therefore the relationship of the norms is In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X - ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

N_L/K(γ) = N(γ)^d.

The norm of an algebraic integer is again an integer. In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...

In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is an ideal of O_K, the ring of integers of the number field K, N(I) is the number of residue classes in O_K/I - i.e. the cardinality of this finite ring. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αO_K there is the expected relation between N(I) and the absolute value of the norm to Q of α, for α an algebraic integer. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like even number or multiple of 3. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called the degree... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ...

See also: field trace. In mathematics, the field trace is a linear mapping defined for certain field extensions. ...