# Ticket #1359(needs_info enhancement)

Opened 5 years ago

## implement cyclotomic norm residues

Reported by: Owned by: was davidloeffler major sage-5.10 number fields N/A

### Description

Here's the code, basically:

bug day 6 -- #1342 system:sage

```K.<zeta> = CyclotomicField(7)
```
```def norm_symbol_prime(a, P):
K = P.number_field()
zeta = K.gen()
n = K.zeta_order()
exponent = (1/n) * ( P.norm() - 1)
exponent = ZZ(exponent)
FF = K.residue_field(P)
aa = FF(a)
b = FF(a)^exponent
zeta_mod = FF(zeta)
# Find power m of zeta_mod that is equal to b, then
# return zeta^m
m = 0
w = FF(1)
while w != b:
w = w * zeta_mod
m += 1
assert m <= n, "bug in norm_symbol_prime"
return zeta^m

def norm_symbol(a, b):
F = K.fractional_ideal([b]).factor()
return prod([norm_symbol_prime(a, P)^e for P, e in F],
K(1))
```
```norm_symbol(zeta^3, 13*zeta)
///
-zeta^5 - zeta^4 - zeta^3 - zeta^2 - zeta - 1
```
```norm_symbol(zeta^7, K(11))
///
1
```
```norm_symbol((1+zeta)^2, 23*zeta)
///
zeta^4
```

## Change History

### comment:1 Changed 4 years ago by davidloeffler

• Owner changed from was to davidloeffler
• Component changed from number theory to number fields

### comment:2 Changed 20 months ago by jdemeyer

• Status changed from new to needs_info
• Report Upstream set to N/A

Please explain what are "cyclotomic norm residues".

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