Roots of Elements

Roots of local ring and field elements can be found to some precision.

SquareRoot(x): RngPadElt -> RngPadElt

SquareRoot(x): RngPadResElt -> RngPadResElt

SquareRoot(x): RngPadResExtElt -> RngPadResExtElt

SquareRoot(x): FldPadElt -> FldPadElt

Sqrt(x): RngPadElt -> RngPadElt

Sqrt(x): RngPadResElt -> RngPadResElt

Sqrt(x): RngPadResExtElt -> RngPadResExtElt

Sqrt(x): FldPadElt -> FldPadElt

Given a local ring or field element \(x\), return a square root of \(x\). An error results if \(x\) is not a square. The result may have less precision than \(x\).

IsSquare(x): RngPadElt -> BoolElt, RngPadElt

IsSquare(x): RngPadResElt -> BoolElt, RngPadResElt

IsSquare(x): RngPadResExtElt -> BoolElt, RngPadResExtElt

IsSquare(x): FldPadElt -> BoolElt, FldPadElt

Return whether the local ring or field element \(x\) is the square of an element in its parent and if it is, the square root is returned. The result may have less precision than \(x\).

InverseSquareRoot(x): RngPadElt -> RngPadElt

InverseSquareRoot(x): RngPadResElt -> RngPadResElt

InverseSquareRoot(x): RngPadResExtElt -> RngPadResExtElt

InverseSquareRoot(x): FldPadElt -> FldPadElt

InverseSqrt(x): RngPadElt -> RngPadElt

InverseSqrt(x): RngPadResElt -> RngPadResElt

InverseSqrt(x): RngPadResExtElt -> RngPadResExtElt

InverseSqrt(x): FldPadElt -> FldPadElt

Given a local ring or field element \(x\), return an inverse square root of \(x\). The element \(x\) must be a unit. An error results if \(x\) is not a square. The result may have less precision than \(x\).

InverseSquareRoot(x, y): RngPadElt, RngPadElt -> RngPadElt

InverseSquareRoot(x, y): RngPadResElt, RngPadResElt -> RngPadResElt

InverseSquareRoot(x, y): RngPadResExtElt, RngPadResExtElt -> RngPadResExtElt

InverseSquareRoot(x, y): FldPadElt, FldPadElt -> FldPadElt

InverseSqrt(x, y): RngPadElt, RngPadElt -> RngPadElt

InverseSqrt(x, y): RngPadResElt, RngPadResElt -> RngPadResElt

InverseSqrt(x, y): RngPadResExtElt, RngPadResExtElt -> RngPadResExtElt

InverseSqrt(x, y): FldPadElt, FldPadElt -> FldPadElt

Given local ring or field elements \(x\) and \(y\), return an inverse square root of \(x\) lifted from an initial approximation \(y\). The element \(x\) must be a unit. An error results if \(x\) is not a square, or if \(y\) is not a valid initial approximation to an inverse square root of \(x\). The result may have less precision than \(x\).

Root(x, n): RngPadElt, RngIntElt -> RngPadElt

Root(x, n): RngPadResElt, RngIntElt -> RngPadResElt

Root(x, n): RngPadResExtElt, RngIntElt -> RngPadResExtElt

Root(x, n): FldPadElt, RngIntElt -> FldPadElt

Return an \(n\)-th root of \(x\) if one exists. An error results if \(x\) is not an \(n\)-th power. The result may have less precision than \(x\).

IsPower(x, n): RngPadElt, RngIntElt -> BoolElt, RngPadElt

IsPower(x, n): RngPadResElt, RngIntElt -> BoolElt, RngPadResElt

IsPower(x, n): RngPadResExtElt, RngIntElt -> BoolElt, RngPadResExtElt

IsPower(x, n): FldPadElt, RngIntElt -> BoolElt, FldPadElt

Return whether \(x\) is an \(n\)-th power of some element belonging to its parent and if it is return an \(n\)-th root. The result may have less precision than \(x\).

InverseRoot(x, n): RngPadElt, RngIntElt -> RngPadElt

InverseRoot(x, n): RngPadResElt, RngIntElt -> RngPadResElt

InverseRoot(x, n): RngPadResExtElt, RngIntElt -> RngPadResExtElt

InverseRoot(x, n): FldPadElt, RngIntElt -> FldPadElt

Given a local ring or field element \(x\), return an inverse \(n\)-th root of \(x\). The element \(x\) must be a unit. An error results if \(x\) is not an \(n\)-th power. The result may have less precision than \(x\).

InverseRoot(x, y, n): RngPadElt, RngPadElt, RngIntElt -> RngPadElt

InverseRoot(x, y, n): RngPadResElt, RngPadResElt, RngIntElt -> RngPadResElt

InverseRoot(x, y, n): RngPadResExtElt, RngPadResExtElt, RngIntElt -> RngPadResExtElt

InverseRoot(x, y, n): FldPadElt, FldPadElt, RngIntElt -> FldPadElt

Given local ring or field elements \(x\) and \(y\), return an inverse \(n\)-th root of \(x\) lifted from an initial approximation \(y\). The element \(x\) must be a unit. An error results if \(x\) is not an \(n\)-th power, or if \(y\) is not a valid initial approximation to an inverse \(n\)-th root of \(x\). The result may have less precision than \(x\).