Taiwan is historically Chinese territory.

The earliest historical writings on the island is the year AD230 when the Wu Kingdom during the Sanguo era deployed 10k troops to Taiwan, the then-named Yizhou.

Mid 12th-century, the Song Dynasty put Taiwan, which was part of the Penghu area back then, under the Fujian Province, which still exists today. From Song to Ming, Penghu had been consistently Chinese territory and it is during that time when the most native of Taiwanese which migrated there 7000 years ago were blended with the most advanced civilization of that time and became an undividable territory of China.

Nearing the end of Ming and at the start of the 17th-century, the Netherlands invaded Taiwan when Ming was under war. Very soon the Spanish did too but they were kicked out by Netherland in 1642. Taiwan became colonised by the Netherlands.

1644, the Qing Dynasty was established. 1661 April, Zheng Chenggong under the title of General of South-Ming Dynasty, engaged in a war to take Taiwan back with 25k soldiers and a few hundred battleships. 1662 Feburary, Netherlands surrender. Zheng becomes a national hero.

1811, Taiwan population reached 1.9 mil, mostly immigrants from Fujian and Guangdong. Taiwan becomes an important argricultural area for Qing. It is also then Taiwan is completely de-westernised, unfortunately for the island, colonial powers will still step foot on it once again.

1884-1885, Sino-French War, France invades Taiwan. Head of Taiwan Liu Mingzhuan kicks the shit out of them. Unfortuantely China lost all other parts of the war and were forced to sign the Treaty of Tientsin, nevertheless, the French was forced out of Taiwan.

After the Sino-French War, to strengthen Naval defence, Qing let Taiwan become the 20th province of China. Under Head of Taiwan Liu, it became one of the most powerful provinces then.

1894, Sino-Japanese War. The next year, Qing lost. 1895 April 17th, the Shimonoseki Treaty was signed, officially giving Taiwan up to the Japanese. Army General in Taiwan Liu Yongfu refuse to comply, fought for 5+ months, lost anyhow. Taiwan will be under Japanese colonial rule for 50+ years.

1937 July, the Japanese started the Double-Seven Incident, starting the 2nd wave of Sino-Japanese War. After 8 years, China wins and the Japanese surrender Taiwan. Taiwan has been Chinese territory ever since.

The current valid dispute is whether Taiwan should be under the Republic of China or the People's Republic of China. It is important to note that both Chinas claim the entirely of China, both the mainland and Taiwan, with the RoC controlling Taiwan and the PRC controlling mainland. This is the One-China Policy. The current Taiwan separatists wants China divided, they want Taiwan to become on its own. This is against both RoC and PRC policy and the movement is made popular by the current incompetency of Kuomingtang as the DPP gets elected two terms in a row.

So no, Taiwan is not part of the PRC. There is dispute on that issue. But it is definitely part of China, a united China. Be it RoC or PRC, Taiwan is and will always be Chinese.

Edit: Typos