Baghdad, Iraq (CNN) -- An American soldier was killed in southern Iraq on Sunday, the U.S. military announced, marking the first U.S. fatality since last week's withdrawal of the last combat brigade from the country.

The military said the soldier was killed in an "indirect fire attack" -- a term used to describe rocket and mortar attacks -- while conducting operations in Basra province.

Also on Sunday, four roadside bombs exploded in various Baghdad neighborhoods, targeting civilians and wounding eight people, police officials said. And two mortar shells hit the capital's heavily fortified Green Zone, but caused no injuries, police said.

The U.S. fatality was the first since Thursday's much-publicized withdrawal of the Army's 4th Stryker Brigade Combat Team. The departure of the 4,000-member unit was seen as the symbolic end of U.S. combat operations in Iraq, though the formal change in mission is set for September 1.

Saturday, the U.S. military announced that its "responsible drawdown of forces" was continuing. The number of U.S. troops remaining in Iraq six and a half years after the 2003 invasion is about 52,000, with another 2,000 slated to leave by September 1.

The remaining Americans are to switch to an advise-and-assist role after that -- "But they certainly have the ability to protect themselves and if necessary to conduct combat operations if it was required," the U.S. commander in Baghdad, Gen. Ray Odierno, told CNN's "State of the Union" on Sunday.

Insurgents still carry out periodic attacks, such as Sunday night's attack on the Green Zone. The district, formally known as the International Zone, houses Iraqi government offices and the U.S. and British embassies and has come under frequent rocket and mortar fire in the past.

CNN's Jomana Karadsheh and Mohammed Tawfeeq contributed to this report.