Patricia Crane

2021-12-26

Show that no finite field is algebraically closed

otoplilp1

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$n-1$ elements, that satisfy the equation

${x}^{(n-1)}=1$

Therefore, All the fields elements including zero satisfy the equation

${x}^{n}=x$

So, the equation,

${x}^{n}=x+1$

Has no roots.

Thus, the field is not algebraically closed.

Hence proved.

Statement: Algebraically closed fields must be infinite.

Step 2: Proof:

Suppose a field has n elements which are finite, then its multiplicative group has

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.

Navreaiw

Beginner2021-12-28Added 34 answers

No.

Suppose a finite field has elements$a}_{1},{a}_{2},\dots ,{a}_{n$ . Then the polynomial $P\left(x\right)=(x-{a}_{1})(x-{a}_{2})\dots (x-{a}_{n})+1$ has no roots, because P(x) is always 1 and not 0.

Suppose a finite field has elements

user_27qwe

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

Therefore, algebraically closed fields must be infinite. You can start with a finite field such as

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