If we adjoin a root α of an irreducible polynomial f ∈ K[x] to the field K we get an algebraic extension K(α), which is again a field. We call K the base field of K(α).
By Kronecker's construction, we may identify K(α) with the factor ring K[x]/(f), an identification that also provides a method for computing in these extension fields.
It is important to note that different extensions of the same field are entirely different (and its elements lie in different families), even if mathematically one could be embedded in the other one.
Currently GAP only allows extension fields of fields K, when K itself is not an extension field.
AlgebraicExtension( K, f ) O
constructs an extension L of the field K by one root of the irreducible
polynomial f, using Kronecker's construction. L is a
field whose LeftActingDomain is K. The polynomial f is the
DefiningPolynomial of L and the attribute RootOfDefiningPolynomial
of L holds a root of f in L (see RootOfDefiningPolynomial).
gap> x:=Indeterminate(Rationals,"x");; gap> p:=x^4+3*x^2+1;; gap> e:=AlgebraicExtension(Rationals,p); <field in characteristic 0> gap> IsField(e); true gap> a:=RootOfDefiningPolynomial(e); (a)
IsAlgebraicExtension( obj ) C
is the category of algebraic extensions of fields.
gap> IsAlgebraicExtension(e); true gap> IsAlgebraicExtension(Rationals); false
According to Kronecker's construction, the elements of an algebraic extension considered to be polynomials in the primitive element. The elements of the base field are represented as polynomials of degree 0. GAP therefore displays elements of an algebraic extension as polynomials in an indeterminate ``a'', which is a root of the defining polynomial of the extension. Polynomials of degree 0 are displayed with a leading exclamation mark to indicate that they are different from elements of the base field.
The usual field operations are applicable to algebraic elements.
gap> a^3/(a^2+a+1); (-1/2*a+1/2*a^2-1/2*a^3) gap> a*(1/a); !1
The external representation of algebraic extension elements are the
polynomial coefficients in the primitive element a, the operations
ExtRepOfObj and ObjByExtRep can be used for conversion.
gap> ExtRepOfObj(One(a)); [ 1, 0, 0, 0 ] gap> ExtRepOfObj(a^3+2*a-9); [ -9, 2, 0, 1 ] gap> ObjByExtRep(FamilyObj(a),[3,19,-27,433]); (3+19*a-27*a^2+433*a^3)
GAP does not embed the base field in its algebraic extensions and
therefore lists which contain elements of the base field and of the
extension are not homogeneous and thus cannot be used as polynomial
coefficients or to form matrices. The remedy is to multiply the
list(s) with the One of the extension which will embed all entries in the
extension.
gap> m:=[[1,a],[0,1]]; [ [ 1, (a) ], [ 0, 1 ] ] gap> IsMatrix(m); false gap> m:=m*One(e); [ [ !1, (a) ], [ !0, !1 ] ] gap> IsMatrix(m); true gap> m^2; [ [ !1, (2*a) ], [ !0, !1 ] ]
IsAlgebraicElement( obj ) C
is the category for elements of an algebraic extension.
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GAP 4 manual
March 2006