Show that no finite field is algebraically closed

Patricia Crane

Patricia Crane

Answered question


Show that no finite field is algebraically closed

Answer & Explanation



Beginner2021-12-27Added 41 answers

Step 1: Given to Show no finite field is algebraically closed.
Statement: Algebraically closed fields must be infinite.
Step 2: Proof:
Suppose a field has n elements which are finite, then its multiplicative group has n1 elements, that satisfy the equation
Therefore, All the fields elements including zero satisfy the equation
So, the equation,
Has no roots.
Thus, the field is not algebraically closed.
Hence proved.


Beginner2021-12-28Added 34 answers

Suppose a finite field has elements a1,a2,,an. Then the polynomial P(x)=(xa1)(xa2)(xan)+1 has no roots, because P(x) is always 1 and not 0.


Skilled2022-01-05Added 375 answers

Nope. An algebraically closed field must, for example, contain all roots of unity of all orders: the roots of the polynomial XN1 . For any N not divisible by the characteristic, this polynomial must have N distinct roots.
Therefore, algebraically closed fields must be infinite. You can start with a finite field such as Fp , the field of residues modulo a prime p, and then you can take an algebraic closure of it Fp. You’ll get a wonderful algebraically closed field of characteristic p, but it’s infinite.

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