The 72-year-old Italian guided his team into the finals of
However, his existing contract is due to expire at the end of the campaign, and talks have yet to open over an extension.
Andrews, who scored the opening goal in the 4-0 first leg win in
He said: "It would be ludicrous if they didn't try to build on this. Good things don't happen overnight.
"He has come in and he has built a system and a squad, a team with real togetherness and a formula that has become very successful.
"They would be mad not to try to keep him."
Midfielder Andrews, currently on loan at Ipswich from Blackburn, has been a major beneficiary of the system Trapattoni has chosen as the one he believes makes best use of the players at his disposal.
However, the spirit fostered within the camp has proved almost as important during the two years since
Andrews said: "Since Paris, we have stuck together as a group through thick and thin. Obviously, that was a low point in all of our careers, but since then, we have done quite well.
"We have grown as a group and become closer-knit, if that was even possible, and topping it all off with this achievement is fantastic.
"We are a team. Anyone who sees us play or sees us train or sees us in the hotel, it's fairly obvious that there are no prima donnas in this team, there are no egos.
"We love joining up and getting together, and we thoroughly enjoy each other's company, and that's fairly clear to see."
The Republic will discover what lies ahead for it next summer when the draw is made in
In the meantime, it will attempt to take in exactly what it has achieved over an eventful 16-month period which culminated in a remarkable victory at the A Le Coq Arena and an emotional night at the
Andrews said: "It's mental. It hasn't sunk in yet.
"Obviously, over the weekend, we had a good feeling that we would be there, but to finally be there now and to look forward to the draw on
Ireland Republic
Given one of many injury problems for Ireland
Republic of Ireland goalkeeper Shay Given is to fly to London to consult a knee specialist amid mounting injury concerns for manager Giovanni Trapattoni.
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