To prove that \( 1x = x \) for all \( x \) in a finite field \( \mathbb{F}_n \), we can use the basic properties of fields. Here's a step-by-step proof:

### **Proof:**

1. **Field Axioms:**
   By definition, a field \( \mathbb{F}_n \) is a set equipped with two operations, addition (+) and multiplication (·), that satisfy certain axioms (e.g., commutativity, associativity, distributivity, existence of identity elements, and existence of inverses for all elements except zero for multiplication).

2. **Multiplicative Identity:**
   One of the axioms of a field is the existence of a multiplicative identity, denoted as 1, such that for any element \( x \in \mathbb{F}_n \),
   \[
   1 \cdot x = x \cdot 1 = x
   \]
   This property holds for all \( x \) in the field \( \mathbb{F}_n \).

3. **Applying the Axiom:**
   According to this axiom, by multiplying any element \( x \) by 1, the result is \( x \) itself. Specifically:
   \[
   1 \cdot x = x
   \]
   This shows that the element 1 acts as the multiplicative identity, meaning it leaves any element \( x \) unchanged when multiplied by it.

4. **Conclusion:**
   Therefore, for any \( x \) in the field \( \mathbb{F}_n \), we have shown that \( 1x = x \), proving the statement.

This completes the proof that \( 1x = x \) for all \( x \) in \( \mathbb{F}_n \).

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the definition of a finite field \( \mathbb{F}_n \)?
2. Why is the existence of a multiplicative identity important in a field?
3. How can you prove that \( 0 \cdot x = 0 \) in a field \( \mathbb{F}_n \)?
4. What are some examples of finite fields?
5. How does the property \( 1x = x \) relate to the concept of scalar multiplication in vector spaces?

**Tip:** When dealing with fields, always remember that the identity elements (both additive and multiplicative) play a crucial role in defining the structure and properties of the field.

Understand and prove the statement 1x = x for all x in finite fields. Learn the fundamental properties of fields, including the existence of a multiplicative identity. Suitable for undergraduate students and those interested in field theory and finite fields.

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