The 20-year-old joins the Reds until the end of the season.
The second-year professional has been on loan with Evo-Stik League South side Frickley Athletic, Evo-Stik Premier Matlock Town and Alfreton`s league rivals Gainsborough Trinity over the last two seasons.
Burdett progressed through the whole academy cycle, joining the club at under-9 level and progressing through to being offered a professional deal.
Last season Burdett found himself on loan at Frickley during the 2016/17 campaign where he made fourteen appearances.
A spell at Gainsborough also followed during the same season.
He was then released by Scunthorpe this month.
Alfreton manager John McDermott told his club`s official website: “I was aware of the player and he has had some spells at different clubs including Gainsborough and Harrogate.
“When I was at the club it was difficult for him to break into the team at the time. But he always worked hard in training; he’s a player that I like, and I’ve been trying to bring players in of a certain style and you have to be patient.
“It was never going to be an overnight transformation here at the Reds and we’ve had to do a lot of business to get the right players in for the system and we’ve had to let players leave in order to get the right ones in that will suit the style.
“Noel has great ball retention, can play number 10 if needed and is a clever player. He’s tidy, creative and can travel well with the ball. He likes to beat a man and it’s excellent to have him with us, he’s had full time fitness at a professional club and we had a chat and his mindset and style of play suits ours. It’s great to have him here”.