English: Scan via Heritage Auctions. Cropped from the original image by uploader.

English: Jacket design by Al Nagy. Published by Doubleday.

English: No permission is required for two reasons:

1. First, the book cover design is too simple to meet the threshold of originality under US copyright law. The design consists of basic text elements and a simple gradient combination of colors. See also: Commons:Threshold of originality § Partial copying or cropping of copyrighted works.
2. Second, the dust jacket was first published prior to 1978 without a valid copyright notice and is ineligible for copyright protection anyway, regardless of the design's simplicity.

Roots was first published in 1976. The hardback book itself carried a valid copyright notice, and its contents remain copyrighted. However, the first-edition dust jacket did not carry a separate copyright notice. According to The Compendium of U.S. Copyright Office Practices: Chapter 2200, § 2207.1(C) at p. 15:

A notice of copyright on the dust jacket of a book is not an acceptable notice for the book, because the dust jacket is not permanently attached to the book. Likewise, a notice appearing in a book is not an acceptable notice for the dust jacket or any material appearing on that dust jacket, even if the book refers to the jacket or material appearing on the jacket.

Keep in mind that the pre-1989 requirements for copyright notice were highly formalistic and, other than a few enumerated exceptions, required these three elements:

1. "The symbol © or the word 'Copyright' or the abbreviation 'Copr.' or an acceptable variant such as "(c)";
2. "The year of first publication for the work"; and
3. "The name of the copyright owner, or an abbreviation by which the name can be recognized, or a generally known alternative designation of the owner."

If just one of these elements is omitted, the work is deemed to be published without notice and is not eligible for copyright protection. Neither the year of publication nor a copyright symbol (or any acceptable variant) appear anywhere on the dust jacket. The mere identifications of the publisher, author, or jacket designer do not meet these requirements. The lack of notice on the dust jacket can be verified by examining scans at the following two sites: themodernfirst.com (providing almost the entire jacket except for the lower portion of the front flap) and invaluable.com (providing the entire front flap).