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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=F_{4} 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 **F**_{11}? Give your reply by an integer between 0 et 10.

### Inverses in F13

What is the inverse of the element of the field **F**_{13}? Give your reply by an integer between 0 et 12.

### Inverses in F17

What is the inverse of the element of the field **F**_{17}? Give your reply by an integer between 0 et 16.

### Inverses in F19

What is the inverse of the element of the field **F**_{19}? Give your reply by an integer between 0 et 18.

### Inverses in F5

What is the inverse of the element of the field **F**_{5}? Give your reply by an integer between 0 et 4.

### Inverses in F7

What is the inverse of the element of the field **F**_{7}? Give your reply by an integer between 0 et 6.

### Element order over F11

What is the multiplicative order of the element of the field **F**_{11} ?

### Element order over F13

What is the multiplicative order of the element of the field **F**_{13} ?

### Element order over F16

Let x be an element of the field **F**_{16} 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 **F**_{17} ?

### Element order over F19

What is the multiplicative order of the element of the field **F**_{19} ?

### Element order over F25

Let x be an element of the field **F**_{25} such that =0. What is the multiplicative order of x?

### Element order over F27

Let x be an element of the field **F**_{27} 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 **F**_{7} ?

### Element order over F9

Let x be an element of the field **F**_{9} such that =0. What is the multiplicative order of x?

### Primitive power over F16

The polynomial P(x)= is irreducible and primitive over **F**_{2}, therefore if r is a root of P(x) in the field K=**F**_{16}, any non-zero element s of K can be written as s=r^{n} for a certain power n. What is the n such that = r^{n} ? (Give your answer by an n between 1 and 15.)

### Primitive power over F8

The polynomial P(x)= is irreducible and primitive over **F**_{2}, therefore if r is a root of P(x) in the field K=**F**_{8}, any non-zero element s of K can be written as s=r^{n} for a certain power n. What is the n such that = r^{n} ? (Give your answer by an n between 1 and 7.)

### Primitive power over F9

The polynomial P(x)= is irreducible and primitive over **F**_{3}, therefore if r is a root of P(x) in the field K=**F**_{9}, any non-zero element s of K can be written as s=r^{n} for a certain power n. What is the n such that = r^{n} ? (Give your answer by an n between 1 and 8.)

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