Polycarp, Clement, and Ignatius were the students of the original disciples. They lived at the turn of the century, before and after 100 AD. They did not mention a trinity or give a description of a trinity in all their writings.
It was not until the second century AD that the idea of a trinity began to take shape in the Gnostic Christian community.
Quintus Septimius Florens Tertullianus is also known as Tertullian. In the second century he formulated the idea of one substance having three persons.
Origen considered the Son to be not coequal, but derived from the Father whom is the Holy Spirit. Arius would adopt the idea of the Son being derivative of the Father in the third century AD. This eventually lead to a major crisis in the Counsel of Nicea.
Arius gained many followers as he taught that Christ was a created being, created by the Father.
Arians were the followers of Arius.
The creed that came out of the Counsel of Nicea in 325 AD did not explicate the Trinity. It simply proclaimed the divinity of Christ, rejecting Arianism.
There was no resolution on who the Holy Spirit is. That notion would not arise again until the Counsel of Constantinople in 381 AD.
Basilius, also known as Basil, bishop of Caesarea. In the later 3rd century AD, formulated ideas as to what the Holy Spirit was. This was mainly in reaction to Arius who was his enemy doctrinally. Basil and others such as Gregory of Nazianzus and Gregory of Nyssa were encouraged to develop ideas to combat the idea of Arianism. The person who encouraged them was Athanasius who hated Arianism and wanted it done away with.
Philo introduced the idea of a trinity to the Hellenistic Jews of Alexandria.
Philo did not equate the three members of his trinity. He wrote that the middle person of the three, was Yahweh, the Father of the Universe, who is uncreated and unbegotten. God, the Father of the Universe was accompanied by two body-guards: the creative power and the royal power. God being greater than them. These ideas of Philo made a great impact on Christianity.
More on How the Trinity Doctrine Entered Christianity
God commanded, You will not make wrongful use of the name of the LORD your God, for the LORD will not acquit anyone who misuses his name. (Exodus 20:7 NRSV) Because of this command the translators of the Septuagint, did not transliterate the name Yahweh into Greek. They believed that the transliteration would have been a misuse of Gods name. Instead, they translated it as Kyrios, which in English is the word LORD. So therefore, the word Lord/Kyrios, became the name of Yahweh in Greek. It was a common title for masters or men of authority. Also, the New Testament writers applied it to Jesus. In the end, Jesus and God shared the same name title: Lord/Kyrios.
In the early 4th century, Lactantius (born 240 A.D. died 320 A.D.) wrote: He {Jesus} taught that God is one {person} and that He {the Father} alone ought to be adored, nor did He {Jesus} ever call himself God. Lactantius did not recognize a Trinity. He emphasized that Jesus is an improperly called god, and must not be worshipped as God.
Wrong interpretations and the distortion of Gods word is what supports the doctrine of the trinity.
When asked, "Which is the most important commandment of all?" Jesus answered, "The most important of all the commandments is, hear, o Israel, the Lord our God is One." (Mark 12:29.)
That is what we find throughout the scriptures:
"Beside me there is no God." (Isa. 44:6.)
"I am God, and there is none else; there is no God beside me." (Isa. 45:5.)
"I am God, and there is none else." (Isa. 46:9.)
"One God and Father of all, who is above all." (Eph. 4:6.)
"Hear, o Israel, the Lord our God is one Lord." (Deut. 6.4.)
"There is one God, and one mediator between God and men, the man Christ Jesus." (1 Tim. 2:5.)
"There is but one God, the Father, whom made all things, and us by Himself , and one lord Jesus Christ, by whom we are in.
(1 Cor. 8:6.)
"This is life eternal, that they might know you, the only true God, and Jesus Christ, whom He has sent." (John 17:3.)
This last quotation is Jesus speaking; addressing God in prayer as the one true God, and speaking of himself as separate from that one true God.
Additional
Ps 110:1
1 The LORD said unto my Lord, Sit Thou at My right hand, until I make Thine enemies Thy footstool.
(KJV)
Acts 2:34-35
34 For David is not ascended into the heavens: but he saith himself, The LORD said unto my Lord, sit thou on my right hand,
35 Until I make thy foes thy footstool.
(KJV)
1 Cor 15:28
28 And when all things shall be subdued unto Him, then shall the Son also himself be subject unto Him that put all things under Him, that God may be all, in all.
(KJV)
Heb 1:2-8
2 Hath in these last days spoken unto us by his Son, Whom He hath appointed heir of all things, by whom also He made the worlds;
3 Who being the brightness of His glory, and the express image of His person, and upholding all things by the word of His power, when He had by Himself purged our sins, sat down on the right hand of the Majesty on high;
4 Being made so much better than the angels, as He hath by inheritance obtained a more excellent name than they.
5 For unto which of the angels said He at any time, "Thou art My Son, this day have I begotten thee? And again, I will be to Him a Father, and He shall be to me a Son?"
(New International Version)
Psalm 97
1 The LORD reigns, let the earth be glad;
let the distant shores rejoice.
2 Clouds and thick darkness surround him;
righteousness and justice are the foundation of his throne.
3 Fire goes before him
and consumes his foes on every side.
4 His lightning lights up the world;
the earth sees and trembles.
5 The mountains melt like wax before the LORD,
before the Lord of all the earth.
6 The heavens proclaim his righteousness,
and all the peoples see his glory.
7 All who worship images are put to shame,
those who boast in idols
worship him, all you gods!
John 17:21
That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.
John 17:23
I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.
Isaiah 44:6
"This is what [the LORD / Yahwah] says Israel's King and Redeemer, [the LORD / God] Almighty: I am the first and I am the last; apart from me there is no [God / Elohiym.]
Isaiah 48:11-12
11 For my own sake, for my own sake, I do this. How can I let myself be defamed? I will not yield my glory to another."Listen to me, O Jacob, Israel, whom I have called: I am he; I am the first and I am the last.
Because KJV Rev 1:11 was found to be a corruption it is not included into other bibles.
KJV Revelation 1:11. Saying, I am Alpha and Omega, the first and the last:
According to history Trinitarianism has its origins in Gnosticism.
There is also a document about a Kabbalist who suggested the Godhead is triune. That was at the same time Yahshua was preaching. Perhaps that is why Yahshua quoted this verse:
Mark 12:29.
"The most important one," answered Jesus, "is this: 'Hear, O Israel, the Lord our God, the Lord is [one / only.]
Trinity
In the fourth-century, Marcellus of Ancyra declared that the idea of the Godhead existing as three hypostases came from Plato, through the teachings of Valentinus. Valentinus is quoted as teaching that God is three, three prosopa (persons) called the Father, the Son and the Holy Spirit:
These men also taught three hypostases, just as Valentinus the heresiarch first invented in the book entitled by him 'On the Three Natures'. It was believed he was the first to invent three hypostases and three persons of the Father, Son and Holy Spirit, but he was discovered to have taken this from Hermes and Plato.
Valentinus (also spelled Valentinius) (c.100 - c.160) was known as a early Christian Gnostic Theologian.
It should be noted that Nag Hammadi library Sethian text such as Trimorphic Protennoia identify Gnosticism as also professing Father, Son and feminine wisdom Sophia or as Professor John D Turner denotes, God the Father, Sophia the Mother, and Logos the Son.