How to find the roots of x^4+1

cofak48

cofak48

Open question

2022-08-15

How to find the roots of x 4 + 1

Answer & Explanation

uiteensitnh

uiteensitnh

Beginner2022-08-16Added 9 answers

Using this, x 4 = 1 = e i π = e ( 2 n + 1 ) π i as e 2 m π i = 1 where m , n are integers.
So, x = e ( 2 n + 1 ) π i 4 = cos ( 2 n + 1 ) π 4 + i sin ( 2 n + 1 ) π 4 where n has any 4 in-congruent values ( mod 4 ), the most simple set of values of n can be { 0 , 1 , 2 , 3 }
If x m = e ( 2 m + 1 ) π i 4 , x m + 2 = e ( 2 m + 3 ) π i 4 = e ( 2 m + 1 ) π i 4 e i π 2 = x m
Also, observe that if y is a solution of x 4 = 1, so is -y
x 0 = cos π 4 + i sin π 4 = 1 + i 2
x 1 = cos 3 π 4 + i sin 3 π 4 = 1 + i 2
x 2 = x 0
x 3 = x 1
So, the values of x are ± ( 1 + i 2 ) , ± ( 1 + i 2 )

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