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 \(x\).