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# RingElement ExtElement -- multiplication of a field element and a Ext-algebra element

## Synopsis

• Operator: SPACE
• Usage:
y = a x
• Inputs:
• a, , $a$ is an element in the field of $E$, where $E$ is of type ExtAlgebra
• x, an instance of the type ExtElement, $x$ is of type $E$
• Outputs:
• y, an instance of the type ExtElement, $y$ is of type $E$, the product $a*x$

## Description

The symbol SPACE is used as notation for multiplication by scalars. The scalars belong to the field of $E$. If the field is not QQ, then the scalars are of type RingElement. If the field is QQ, then the scalars are of type Number. Observe that it is necessary to use the function toField when $F$ is defined as an algebraic extension of a prime field.

 i1 : F = toField(ZZ/7[x]/{x^2+1}) o1 = F o1 : PolynomialRing i2 : L = lieAlgebra({a,b,c},Field=>F)/{a b,b c} o2 = L o2 : LieAlgebra i3 : E = extAlgebra(3,L) o3 = E o3 : ExtAlgebra i4 : (3*x+1) (ext_1 ext_2)+(2*x+3) (ext_2 ext_1) o4 = (-x+2)ext_3 o4 : E