Não, a PRIMARY KEY precisa ser definida na tabela.
Você pode estar pensando isto , que se aplica ao mecanismo InnoDB:
Abaixo está um exemplo que mostra a criação deste índice para uma tabela sem PRIMARY KEY e sem coluna UNIQUE.
# Create the table
create table test.check_table (id int, description varchar(10)) ENGINE = INNODB;
# Verify that there is no primary or unique column
desc test.check_table;
+-------------+-------------+------+-----+---------+-------+
| Field | Type | Null | Key | Default | Extra |
+-------------+-------------+------+-----+---------+-------+
| id | int(11) | YES | | NULL | |
| description | varchar(10) | YES | | NULL | |
+-------------+-------------+------+-----+---------+-------+
# Insert some values
insert into test.check_table values(1, 'value-1');
insert into test.check_table values(2, 'value-2');
insert into test.check_table values(null, 'value-3');
insert into test.check_table values(4, null);
insert into test.check_table values(1, 'value-1');
# Verify table
select * from test.check_table;
+------+-------------+
| id | description |
+------+-------------+
| 1 | value-1 |
| 2 | value-2 |
| NULL | value-3 |
| 4 | NULL |
| 1 | value-1 |
+------+-------------+
# Verify that the GEN_CLUST_INDEX index is auto-created.
select * from INFORMATION_SCHEMA.INNODB_INDEX_STATS where TABLE_SCHEMA='test' and TABLE_NAME = 'check_table';
+--------------+-------------+-----------------+--------+--------------+-------------------+------------------+
| table_schema | table_name | index_name | fields | rows_per_key | index_total_pages | index_leaf_pages |
+--------------+-------------+-----------------+--------+--------------+-------------------+------------------+
| test | check_table | GEN_CLUST_INDEX | 1 | 5 | 1 | 1 |
+--------------+-------------+-----------------+--------+--------------+-------------------+------------------+
# Duplicate rows are still allowed (Primary Key constraints not enforced)
insert into test.check_table values(1, 'value-1');
select * from test.check_table;
+------+-------------+
| id | description |
+------+-------------+
| 1 | value-1 |
| 2 | value-2 |
| NULL | value-3 |
| 4 | NULL |
| 1 | value-1 |
| 5 | value-5 |
| 1 | value-1 |
+------+-------------+
Para contrastar, uma tabela com uma PRIMARY KEY especificada cria um índice com o nome PRIMARY.
# Create another table
create table test.check_table_2 (id int, description varchar(10), PRIMARY KEY(id)) ENGINE = INNODB;
# Verify primary key column
desc check_table_2;
+-------------+-------------+------+-----+---------+-------+
| Field | Type | Null | Key | Default | Extra |
+-------------+-------------+------+-----+---------+-------+
| id | int(11) | NO | PRI | 0 | |
| description | varchar(10) | YES | | NULL | |
+-------------+-------------+------+-----+---------+-------+
# Verify index
select * from INFORMATION_SCHEMA.INNODB_INDEX_STATS where TABLE_SCHEMA='test' and TABLE_NAME = 'check_table_2';
+--------------+---------------+------------+--------+--------------+-------------------+------------------+
| table_schema | table_name | index_name | fields | rows_per_key | index_total_pages | index_leaf_pages |
+--------------+---------------+------------+--------+--------------+-------------------+------------------+
| test | check_table_2 | PRIMARY | 1 | 0 | 1 | 1 |
+--------------+---------------+------------+--------+--------------+-------------------+------------------+
# Primary key is enforced
insert into check_table_2 values(1,'value-1');
OK
insert into check_table_2 values(1,'value-1');
ERROR 1062 (23000): Duplicate entry '1' for key 'PRIMARY'
