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# normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element

- Usage:
    `normalAutomorphism x`
- Inputs:
  - x, a ring element, a homogeneous normal element
- Outputs:
  - a ring map,

## Description

Let x be a homogeneous element in a noncommutative ring R. If x is normal then x determines a graded ring automorphism f of R by x*a = f(x)*a. This method returns this automorphism as a RingMap.

    i1 : A = QQ<|a,b,c|>

    o1 = A

    o1 : FreeAlgebra

    i2 : I = ideal {a*b+b*a,a*c+c*a,b*c+c*b}

    o2 = ideal (a*b + b*a, a*c + c*a, b*c + c*b)

    o2 : Ideal of A

    i3 : B = A/I
    Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
    Converting to Naive algorithm.

    o3 = B

    o3 : FreeAlgebraQuotient

    i4 : sigma = map(B,B,{b,c,a})

    o4 = map (B, B, {b, c, a})

    o4 : RingMap B <-- B

    i5 : C = oreExtension(B,sigma,w)
    Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
    Converting to Naive algorithm.

    o5 = C

    o5 : FreeAlgebraQuotient

By construction, w is normal, and the normalizing automorphism is sigma extended to C sending w to itself. It follows that therefore w^2 is also normal whose automorphism is the square of sigma extended to C in a similar way. We verify these facts with the following commands:

    i6 : isNormal w^2

    o6 = true

    i7 : phi = normalAutomorphism w^2

    o7 = map (C, C, {c, a, b, w})

    o7 : RingMap C <-- C

    i8 : matrix phi

    o8 = | c a b w |

                 1      4
    o8 : Matrix C  <-- C

    i9 : matrix (sigma * sigma)

    o9 = | c a b |

                 1      3
    o9 : Matrix B  <-- B

## See also

- normalElements -- Finds normal elements

## Ways to use normalAutomorphism:

- `normalAutomorphism(RingElement)`

## For the programmer

The object normalAutomorphism is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2:158:0.