Add simple auth

backend/src/main/java/com/aiteacher/auth/AuthController.java

@@ -0,0 +1,19 @@
+package com.aiteacher.auth;
+
+import org.springframework.http.ResponseEntity;
+import org.springframework.web.bind.annotation.GetMapping;
+import org.springframework.web.bind.annotation.RequestMapping;
+import org.springframework.web.bind.annotation.RestController;
+
+import java.security.Principal;
+import java.util.Map;
+
+@RestController
+@RequestMapping("/api/v1/auth")
+public class AuthController {
+
+    @GetMapping("/check")
+    public ResponseEntity<Map<String, String>> check(Principal principal) {
+        return ResponseEntity.ok(Map.of("username", principal.getName()));
+    }
+}