Question

Sei \(V\) ein Vektorraum über einem Körper \(K=G F(p)\), dessen Ordnung eine Primzahl \(p\) ist. Zeigen Sie: Jedes von Null verschiedene Element von \(V\) hat in der additiven Gruppe die Ordnung \(p\).

Short answer

Each non-zero element in the vector space has order \( p \) because the cyclic nature of the field ensures that adding an element \( p \) times results in zero.

Step-by-step solution

Understanding the Vector Space

We start by understanding that in a vector space \( V \) over a field \( K = GF(p) \), each nonzero element of the field \( K \) can be conceptualized as a unit vector. Since \( GF(p) \) is a finite field of prime order \( p \), it has exactly \( p \) elements. The set of these elements forms the additive group of \( V \).

Additive Group Properties

The additive group of a field \( GF(p) \) has elements \( \{0, 1, 2, \ldots, p-1\} \). In this group's operations, each element other than zero forms a cyclic group under addition modulo \( p \). This means if you repeatedly add an element to itself, you eventually cycle back to zero.

Finding the Order of a Nonzero Element

For any non-zero element \( a \in GF(p) \), consider the sequence \( a, 2a, 3a, \ldots \) under addition. We want to find the smallest integer \( n \) such that \( na \equiv 0 \mod p \). Since \( GF(p) \) is a field of prime order, any nonzero element \( a eq 0 \) under addition will replicate zero when added to itself \( p \) times.

Showing the Order is \( p \)

Specifically, consider \( np \equiv 0 \mod p \). This equation is always satisfied when \( n = p \), as every integer multiple of \( p \) is congruent to zero modulo \( p \). Thus, \( pa = 0 \), and the order of any non-zero element \( a \) must be \( p \) in the additive group. This is based on the fact there is no smaller positive integer satisfying this condition.

Conclusion

Thus, we conclude that every non-zero vector in the vector space \( V \) over the field \( GF(p) \) has order \( p \) in the additive group. This is due to the properties of cyclic groups in modular arithmetic when the modulus is a prime number.

Key concepts

Vector Space

A vector space, in its simplest form, is a collection of vectors where you can perform two main operations: vector addition and scalar multiplication. Imagine these vectors as arrows on a graph that can stretch, shrink, or change direction, but they always follow specific rules to stay consistent within the space.
In the context of our exercise, the vector space is defined over a finite field, specifically one with a prime order represented by \( GF(p) \). This means our set of scalars, which we use to stretch or shrink our vectors, has a limited number of elements.
More specifically, the field \( GF(p) \) includes precisely \( p \) elements. This constraint shapes how vector addition and scalar multiplication work, making it an exciting realm to consider when studying algebraic structures. A vector space over a finite field with prime order introduces a unique structure that is simpler in some ways and more complex in others.

Additive Group

An additive group is a set paired with an operation—like addition—that meets all the group criteria. These criteria are closure, associativity, identity, and invertibility.
For our vector space \( V \) over the field \( GF(p) \), the concept of an additive group becomes vital. Here, each element in the vector space can be added to another, and the result is still within the space.
The identity element in this group is the zero element, as adding zero to any element leaves it unchanged. Every element also has an inverse, typically its negative, which when added to the original element results in the identity.

• Closure: The sum of any two elements in the group is also an element of the group.
• Associativity: The way elements are grouped in addition does not affect their sum.
• Identity: There exists an element (zero vector) that does not change an element when added to it.
• Invertibility: Each element has an inverse, a counterpart that sums to the identity.

Understanding these properties helps break down how addition works within a vector space.

Prime Order

In mathematics, a structure having a prime order means there are exactly \( p \) elements in that structure, where \( p \) is a prime number. For our finite field \( GF(p) \), this implies several fascinating properties.
First, a prime number of elements ensures that the only divisors are 1 and the number itself, which simplifies many properties and operations within the field.
Moreover, every non-zero element has a multiplicative inverse, a remarkable property of fields with prime order. This effectively means you can "divide" by any non-zero element within this field. Having a field with a prime order fundamentally impacts every operation's nature and is crucial in determining how vector spaces behave when formed over such fields.

Cyclic Group

A cyclic group is a type of group that can be generated by repeatedly applying the group operation to a single element, known as the generator. Every element in the group can be represented as powers of this generator.
In the case of our additive group derived from \( GF(p) \), each non-zero element forms its cyclic subgroup under addition. This means that by repeatedly adding the element to itself, you will cycle through each element of the group until reaching zero again.
This cycling back to zero highlights an essential property of a cyclic group. In our scenario, when the order of a cyclic group is a prime \( p \), the smallest number such that the repeated addition of an element equals zero is exactly \( p \). No smaller positive number will satisfy this, confirming the group's cyclic nature concerning a prime order field.