If a row of carrots went to seed in the vegetable patch, you should discard them. Dig them up, removing all taproots and above-ground stems and leaves. Do not eat the roots. Avoid shaking the pulled up carrot plants too much, as you risk scattering the seeds haphazardly across the vegetable patch. The seeds will sprout next spring wherever they landed and likely be viewed as weeds. Discard the old carrot plants in the compost pile.
If you abandon the vegetable patch and leave the old carrot plants that hold seed heads on their stems, you can expect wildflower carrots to linger in that area. The seeds will drop to the ground and sprout next spring. You may pull up and harvest these young carrot plants to eat. Otherwise, the carrot plants will grow as biennial plants on their own, becoming attractive Queen Anne's lace flowering specimens. A continual collection of rejuvenating year-old roots and newly sprouting seeds will perpetuate in the area for years.
You may snip off the dried seed heads on the old carrot plants after the first fall frost. Allow the seed heads to further dry on a plate in a cool, dry location indoors. The seeds may be sown next year, but the quality of the plants and their roots will not be similar to the parent carrot. These carrot flowers were pollinated by insects, and the resulting seeds possess random genes from any of the individual carrot flowers that supplied pollen. There is no guarantee these seeds will produce plants with ideal, tasty, desirable roots to eat. Keep dry seeds in an airtight jar in the refrigerator for up to two years.
Carrot seed is both edible dry and used as a fatty oil. In homeopathy, carrot seed may be used to expel intestinal gas -- flatulence -- or as a diuretic or menstruation promoter. While difficult to extract, carrot seed oil is sold in health food stores or natural cosmetic shops. The oil may be used as a rejuvenating emollient in skincare. Tea made from dried flowers may alleviate menstrual cramping.