how to get ID of field based on the Max of another field in same table

I have a table named ae_types. It contains three fields that are relevant to my question:

aetId               This is Auto Increment and is the Primary Key
aetProposalType     Text field 5 characters long
aetDaysToWait       Byte data type

aetProposalType and aetDaysToWait are in a unique key so that I am guaranteed that there will never be two aetProposalTypes with the same aetDaysToWait.

The result that I am looking for is to get the aetId of the field with the largest aetDaysToWait for each aetProposalType.

Below is the query that I have come up with to accomplish this, but it seems to me like it is possibly unnecessarily complicated and not very beautiful.

SELECT a开发者_运维知识库e_types.aetId AS lastEmailId, ae_types.aetProposalType
FROM ae_types INNER JOIN 

(SELECT ae_types.aetProposalType, Max(ae_types.aetDaysToWait) AS MaxOfaetDaysToWait 
FROM ae_types GROUP BY ae_types.aetProposalType)  AS ae_maxDaysToWaitByProposalType 

ON (ae_types.aetDaysToWait = ae_maxDaysToWaitByProposalType.MaxOfaetDaysToWait) 
AND (ae_types.aetProposalType = ae_maxDaysToWaitByProposalType.aetProposalType);

What are some alternative solutions and why would they be better?

PS If you have any questions please ask and I will be happy to attempt to provide the answer.

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That's the way I'd do it too.

select a.aetId, a.aetProposalType, a.aetDaysToWait
    from ae_types a
        inner join (select aetProposalType, max(aetDaysToWait) as MaxDays
                    from ae_types
                    group by aetProposalType) sq
            on a.aetProposalType = sq.aetProposalType
                and a.aetDaysToWait = sq.MaxDays