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Discover doesn't allow joint account holders or cosigners on credit card accounts. However, you can apply for the Discover card in your name and add your wife as an authorized user, or both apply for separate Discover cards for individual use.
An authorized user is someone like a friend or family member added to an account without an additional credit check. They'll get a card with their name on it and share charging privileges (up to the primary cardholder's available line of credit).
You can add your wife as an authorized user to your Discover account by logging into the Account Center and then selecting "Manage Authorized Users." Adding a user won't directly impact your credit score. However, the account activity, including payment history and credit utilization, is reported to the credit bureaus for both users. You, as a primary cardholder, will be solely responsible for all the charges on the account, so poor spending or payment habits by an authorized user can negatively impact your credit.
You, in turn, can benefit by accelerating rewards earning, maximizing card perks, and simplifying shared expenses, while your wife gains access to credit and the opportunity to build her credit history.