evaluation - How do I get Mathematica to return a function call unevaluated?

How do I get Mathematica to return a function call (conditionally) unevaluated? I expect this may use the slightly-mysterious Hold function.

As a toy example, suppose I want to define AlgebraicQ such that AlgebraicQ[x] returns True or False when Element[x, Algebraics] evaluates to True or False, but otherwise to returns AlgebraicQ[x], just like the other predicate functions do. (I can't just ask if Element[x, Algebraics] == True, because this is itself unevaluated.)

Edit: The first thing that came to mind didn't work, as you can see:

I had tried this before posting, but on a recommendation I tried again with a fresh kernel (pictured above) with the same results. I also tried

AlgebraicQ[a_] := True /; Element[x, Algebraics]
AlgebraicQ[a_] := False /; ! Element[x, Algebraics]

based on an earlier suggestion but this seems not to work at all.

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based on Szabolcs' answer:

AlgebraicQ[a_] := With[{result = Element[a, Algebraics]},
  result /; MatchQ[result, True | False]]

which tests as expected:

AlgebraicQ /@ {7, Pi, Pi + E}

Out[2]= {True, False, AlgebraicQ[E + Pi]}

Answer

Here's how this can be done:

ClearAll[algebraicQ]
algebraicQ[x_] := Module[{result},
  result = Element[x, Algebraics]; 
  result /; MatchQ[result, True | False]]

The key to these types of problems is usually a special use of Condition inside Block/Module/With which allows sharing localized variables between the condition and the body of Module.

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At this point I should note that the convention seems to be that any function that ends in ...Q will always return either True or False. Consider EvenQ vs Positive. EvenQ[x], with x undefined, gives False. Positive[x] stays unevaluated. I know of only a very few edge cases which don't follow this. Naming this algebraicQ would violate that convention.