  GNU-Darwin Web

### `sp_norm`

sp_norm(alg,var,poly,alglist)
:: Norm computation over an algebraic number field.
return
polynomial
var
The main variable of poly
poly
univariate polynomial
alg
`root`
alglist
`root` list
• Defined in the file `sp'.
• Computes the norm of poly with respect to alg. Namely, if we write K = Q(alglist \ {alg}), The function returns a product of all conjugates of poly, where the conjugate of polynomial poly is a polynomial in which the algebraic number alg is substituted for its conjugate over K.
• The result is a polynomial over K.
• The method of computation depends on the input. Currently direct computation of resultant and Chinese remainder theorem are used but the selection is not necessarily optimal. By setting the global variable `USE_RES` to 1, the builtin function `res()` is always used.
``` load("sp")\$
 A0=newalg(x^2+1)\$
 A1=newalg(x^2+A0)\$
 sp_norm(A1,x,x^3+A0*x+A1,[A1,A0]);
x^6+(2*#0)*x^4+(#0^2)*x^2+(#0)
 sp_norm(A0,x,@@,[A0]);
x^12+2*x^8+5*x^4+1
```
Reference
section `res`, section `asq`, `af`, `af_noalg`