On a laptop, hold down the FN and left ALT keys at the same time and use the FN number keys, not the row of numbers under the Function keys.
On a PC, hold down the left ALT key and use the number pad keys, not the row of numbers under the Function keys.
Key the numbers 0196 for Ä and 0228 for ä.
Ä = 0196
ä = 0228
Ö = 0214
ö = 0246
Ü = 0220
ü = 0252
ß = 0223
€ = 0128
¢ = 0162
© = 0169
® = 0174
™ = 0153
¼ = 0188
½ = 0189
¾ = 0190
° = 0176
÷ = 0247
± = 0177
Ø = 0216
Ω = 8486
This works on my Dell laptop and PC with U.S. keyboards.