How do I find the max of one column with a condition linked to another column

Using the nycflights13 dataset, I want to find, using R, which flight was the latest in each month; in other words, the flight with the largest departure delay in each month.

The code I've used:

flights %>% group_by(flights$month) %>% summarize(largest_delay = max(flights$dep_delay, na.rm=TRUE))

this gives me a table of the months along with the the largest departure delay across the entire dataset, rather than the month-wise maximum:

> flights %>% group_by(flights$month) %>% summarize(largest_delay = max(flights$dep_delay, na.rm=TRUE))
# A tibble: 12 x 2
   flights$month` largest_delay
             <int>         <dbl>
 1               1          1301
 2               2          1301
 3               3          1301
 4               4          1301
 5               5          1301
 6               6          1301
 7               7          1301
 8               8          1301
 9               9          1301
10              10          1301
11              11          1301
12              12          1301

My question: how would I modify the above code such that it gives me the month-wise max? Also, how can I add in an additional column that contains the tailnum corresponding to that flight?