Is it possible to create a button that is the child of an text?
This will result in invalid HTML, so its not possible to do this. What you can do is create a div, which is a block element and fill it with a text box + button Inline elements.
With some styling in your CSS you can make your div look like a textbox and your textbox like a div, without borders and bevels. Resulting in something that looks like the thing you are trying to achieve here.
To get rid of the textbox look on your input use #YourTextBox {border: none;}