1. Understand the question

We need to identify the element that does not show variable oxidation state among:

• Bromine
• Iodine
• Chlorine
• Fluorine

2. Concept used

Halogens generally show oxidation state $-1$. However, except fluorine, other halogens can also show positive oxidation states such as $+1, +3, +5, +7$ in compounds with oxygen or more electronegative atoms.

The reason is:

• Fluorine is the most electronegative element.
• It cannot show positive oxidation states.
• In its compounds, fluorine almost always has oxidation state $-1$.

3. Check each option

A: Bromine

Bromine shows oxidation states like: $-1, +1, +3, +5, +7$ So, bromine does show variable oxidation states.

B: Iodine

Iodine shows oxidation states like: $-1, +1, +3, +5, +7$ So, iodine does show variable oxidation states.

C: Chlorine

Chlorine shows oxidation states like: $-1, +1, +3, +5, +7$ So, chlorine does show variable oxidation states.

D: Fluorine

Fluorine shows only: $-1$ It does not show variable oxidation state.

4. Final answer

Therefore, the element not showing variable oxidation state is: $\boxed{\text{Fluorine}}$

So, the correct option is D.

5. Comparison with stored correct answer

Stored correct answer: D

My derived answer: D

They match.