Question: In which battle did Ukita Hideie fight Tokugawa Ieyasu resulting in Hideie's exile?

Context articles: Tokugawa Ieyasu
Tokugawa Iemitsu
Japanese Left Army
Date Hidemune
Ukita Hideie
Arima Naozumi
Council of Five Elders
Ii Naokatsu
Battle of Byeokjegwan
Akashi Takenori

Tokugawa Ieyasu (徳川 家康 , January 31, 1543 – June 1, 1616) was the founder and first shogun of the Tokugawa shogunate of Japan, which effectively ruled Japan from the Battle of Sekigahara in 1600 until the Meiji Restoration in 1868.
 Ieyasu seized power in 1600, received appointment as shogun in 1603, and abdicated from office in 1605, but remained in power until his death in 1616.
 His given name is sometimes spelled Iyeyasu, according to the historical pronunciation of the kana character "he".
 Ieyasu was posthumously enshrined at Nikkō Tōshō-gū with the name Tōshō Daigongen (東照大権現 ) .
 He was one of the three unifiers of Japan, along with his former lord Nobunaga and Toyotomi Hideyoshi.
Tokugawa Iemitsu (徳川 家光 August 12, 1604 – June 8, 1651) was the third shogun of the Tokugawa dynasty.
 He was the eldest son of Tokugawa Hidetada, and the grandson of Tokugawa Ieyasu.
 Iemitsu ruled from 1623 to 1651, and during this period he crucified Christians, expelled all Europeans from Japan and closed the borders of the country, a foreign politics policy that continued for over 200 years after its institution.
 It is debatable whether Iemitsu can be considered a kinslayer for making his younger brother Tadanaga commit suicide by seppuku.
 Iemitsu also had well-known homosexual preferences, and it is speculated he was the last direct male descendant of Tokugawa Ieyasu, thereby ending the patrilineality of the shogunate by the third generation.
The Japanese Left Army was a Japanese army in the 16th-century Japanese invasions of Korea.
 Under the command of Ukita Hideie in the late 16th century, it consisted mainly of the former First division led by Konishi Yukinaga, the Fourth Division led by Shimazu Yoshihiro and the Eighth Division led by Ukita Hideie.
Date Hidemune (伊達 秀宗 , November 11, 1591 – July 8, 1658) was a Japanese daimyo of the early Edo period.
 He was the eldest son of Date Masamune, born in 1596 by Lady Iisaka (a concubine).
 Coming of age while living with Toyotomi Hideyoshi, he received a character from Hideyoshi's name and took the adult name of Hidemune.
 Hideyoshi also granted him the court rank of junior 5th, lower grade (従五位下 , ju go-i no ge ) and the title of "ji-jū", appointing the young Hidemune as a page to his own son Toyotomi Hideyori.
 After Hideyoshi's death in 1600, he was made a hostage at the residence of Ukita Hideie.
Ukita Hideie (宇喜多 秀家 , 1573 – December 17, 1655) was the "daimyō" of Bizen and Mimasaka Provinces (modern Okayama Prefecture), and one of the council of Five Elders appointed by Toyotomi Hideyoshi.
 Son of Ukita Naoie, he married Gōhime, a daughter of Maeda Toshiie.
 Having fought against Tokugawa Ieyasu in the Battle of Sekigahara he was exiled to the island prison of Hachijōjima, where he died.
Arima Naozumi (有馬 直純 , 1586 – June 3, 1641) was the first son of the Kirishitan "daimyō" Arima Harunobu.
 He was baptized as a child with the name Miguel (ミゲル).
 He was born in Hinoe Castle in Shimabara but was sent by his father to work beside Tokugawa Ieyasu at the age of 15.
 He married Konishi Yukinaga's niece Marta (マルタ); however, in order to curry favor with Ieyasu, he divorced his Christian wife and married Ieyasu's adopted daughter Kuni-hime in 1610.
 In 1612, he inherited his father's land valued at 40,000 "koku" in Shimabara when his father was executed for his role in the Okamoto Daihachi Incident.
 Tokugawa Ieyasu ordered a general persecution of all Christians in Japan, and Naozumi immediately gave up his Christian belief, exiled his former wife and secretly killed his two half brothers: 8-year old Francisco (フランシスコ) and 6-year old Mathias (マティアス).
The council of five elders, also known as the five Tairō (五大老 , go-tairō ) , was formed in 1595 by Toyotomi Hideyoshi to rule Japan in the place of his son, Hideyori, until such time as he came of age.
 Hideyoshi chose his five most powerful "daimyōs": Ukita Hideie, Maeda Toshiie, Uesugi Kagekatsu, Mōri Terumoto, and the famous Tokugawa Ieyasu.
 (Kobayakawa Takakage was also to have been one of the elders, but died before Hideyoshi himself.)
Ii Naokatsu (井伊 直勝 , 1590 – August 24, 1662) was a Japanese daimyo of the Edo period who served the Tokugawa clan.
 He was also known as Ii Naotsugu.
 Naokatsu succeeded to family headship following his father's death in 1602.
 Under Tokugawa Ieyasu's orders, Naokatsu completed construction of Hikone Castle in 1606, and then moved there from Sawayama Castle when it was largely completed.
 In 1614, as Naokatsu was ill, he sent his brother Naotaka to fight in the Siege of Osaka; Naokatsu himself was assigned to Annaka, where he undertook security duty in the Kantō region.
 After the siege of Osaka, Tokugawa Ieyasu rewarded Naokatsu's younger brother Naotaka with the Ii family headship, and allowed Naokatsu to form a branch family with holdings at the fief of Annaka in Kōzuke Province, worth some 30,000-koku.
 Naokatsu retired in 1632, yielding headship to his son Naoyoshi.
 He died in Ōmi Province in 1662.
The Battle of Byeokjegwan (or Pyŏkje) (Chinese: 碧蹄館大戰; "Bì tí guǎn dàzhàn") was a battle fought on January 27, 1593 (January 26 according to the Japanese calendar of the time), between the armies of the Ming Dynasty led by Li Rusong, and the Japanese forces under Tachibana Muneshige, Ukita Hideie, and Kobayakawa Takakage.
 As part of the Japanese Invasion of Korea (Imjin War), it was the first field battle fought during the war between the two sides.
Akashi Takenori (明石 全登 , 1566–1618?)
 was a Japanese samurai of the Azuchi-Momoyama through early Edo periods. Also known as Teruzumi, Zentō, or Naritoyo.
 Retainer of Ukita Naoie, the major daimyō of Bizen Province.
 Also known by his court title, Kamon-no-Kami (掃部頭 ) .
 Takenori also served as a strategist under Naoie's son Ukita Hideie.
 At the Battle of Sekigahara, he fought bravely against Fukushima Masanori.
 After the Ukita clan had been destroyed in the Battle of Sekigahara, Akashi vanished.
 At the Siege of Osaka, Akashi entered the Osaka castle and he fought against Tokugawa Ieyasu to the last minute.
 After the castle's fall, he escaped again.
 He never committed suicide because of his Christian beliefs.
 Despite being hunted by the forces of Tokugawa Ieyasu, he was not caught; his whereabouts after the battle are unknown.