# OEF finite field --- Introduction ---

This module actually contains 21 exercises on finite fields.

### Arithmetics over F4

We designate the 4 elements of the field K=F4 by 0,1,2,3, where 0 and 1 are the respective neutral elements of the additive and multiplicative groups of K.

What is the element in K ?

### Primitive counting

Compute the number of primitive elements in the finite field K=F.

Recall. A non-zero element x in K is primitive, if x is a generator of the multiplicative group of K.

### Element power

Compute the element in the finite field K=F.

### Inverses in F11

What is the inverse of the element of the field F11?

### Inverses in F13

What is the inverse of the element of the field F13?

### Inverses in F17

What is the inverse of the element of the field F17?

### Inverses in F19

What is the inverse of the element of the field F19?

### Inverses in F5

What is the inverse of the element of the field F5?

### Inverses in F7

What is the inverse of the element of the field F7?

### Element order over F11

What is the multiplicative order of the element of the field F11 ?

### Element order over F13

What is the multiplicative order of the element of the field F13 ?

### Element order over F16

Let x be an element of the field F16 such that =0. What is the multiplicative order of x?

### Element order over F17

What is the multiplicative order of the element of the field F17 ?

### Element order over F19

What is the multiplicative order of the element of the field F19 ?

### Element order over F25

Let x be an element of the field F25 such that =0. What is the multiplicative order of x?

### Element order over F27

Let x be an element of the field F27 such that =0. What is the multiplicative order of x?

### Element order over F7

What is the multiplicative order of the element of the field F7 ?

### Element order over F9

Let x be an element of the field F9 such that =0. What is the multiplicative order of x?

### Primitive power over F16

The polynomial P(x)= is irreducible and primitive over F2, therefore if r is a root of P(x) in the field K=F16, any non-zero element s of K can be written as s=rn for a certain power n.

What is the n such that  = rn ? (Give your answer by an n between 1 and 15.)

### Primitive power over F8

The polynomial P(x)= is irreducible and primitive over F2, therefore if r is a root of P(x) in the field K=F8, any non-zero element s of K can be written as s=rn for a certain power n.

What is the n such that  = rn ? (Give your answer by an n between 1 and 7.)

### Primitive power over F9

The polynomial P(x)= is irreducible and primitive over F3, therefore if r is a root of P(x) in the field K=F9, any non-zero element s of K can be written as s=rn for a certain power n.

What is the n such that  = rn ? (Give your answer by an n between 1 and 8.) The most recent version

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