If 'No politician is dishonest' is false, then the statement 'Some politicians are dishonest' shall be

The statement "Some politicians are dishonest" must be true.

"No politician is dishonest" is a universal negative, an E proposition.

"Some politicians are dishonest" is its particular affirmative, an I proposition.

On the square of opposition E and I are contradictories.

Contradictories always take opposite truth-values.

So when the E statement is false, the I statement is necessarily true.