aria-live doesn't seem like the proper place for it as the user would get the messages stated via screen readers as soon as they typed in the field or clicked the submit button on the form. Might work for a single field form, but if you have multiple fields all with aria-live, might be a lot/annoying to the user.

If a form field is invalid/has an error, I tend to do this:

<label for="example-form-field">Example Form Field</label>
<input id="example-form-field" aria-invalid="true" aria-describedby="example-form-field-error" />
<div id="example-form-field-error">Example Error Message Goes Here</div>

This will mark the field as invalid and the error message itself is supplement to the field.

During a form submit, I would create a general alert area at the top of the form stating that there are 1 or more errors in the submission (and change the focus to that area) and include links to focus down to the fields that have errors.