核

Phono-semantic compound (形聲 /形声, OC *ɡuːd, *ɡrɯːɡ): semantic 木 (“tree”) + phonetic 亥 (OC *ɡɯːʔ) – a kind of tree. Modern meanings are borrowed.
The definitions related to “to examine” come from the “kernel” sense, i.e. “to go to the core” (Schuessler, 2007).
Generally thought to be of Sino-Tibetan origin. It has been compared to Tibetan རག་ཙེ (rag tse), Karbi rak (Gong, 2002; Schuessler, 2007). However, Hill (2012) rejects the connection to the Tibetan word because most words ending with ཙེ (tse) are loanwords from Chinese words ending with the 子 suffix.
Smith (2011) groups this word with the following terms into a word family meaning “root, germ, generative core”:
* 荄 (OC *kɯː, “grassroot”) and its derivative 根 (OC *kɯːn, “root”),
* 骸 (OC *ɡrɯː, “(human) bones”) (also noted by Schuessler, 2007),
* 孩 (OC *ɡɯː, “child”),
* 亥 (OC *ɡɯːʔ, “sprouting stage”), i.e. "the moon’s first appearance" > "twelfth earthly branch".

Han compound

(mathematics, algebra, linear algebra) kernel


次

Characters in the same phonetic series (次) (Zhengzhang, 2003)
Ideogrammic compound (會意 /会意) and possibly phono-semantic compound (形聲 /形声, OC *sn̥ʰis): phonetic 二 (OC *njis, “two”) + semantic 欠. Given that the right part of the character depicts a man kneeling with an open mouth, the number two perhaps indicates "two people speaking".
As an alternative, the two horizontal traits 二 are the representation of some saliva spitted while speaking; hence, the act of speaking is represented without a reference to blown air or breathing (e.g., 曰, 丂, 于, 只). The original meaning was "to consult", 咨. Perhaps related to 㳄, which means "saliva"; however, this is not the original version of the character, but a variant.
Given that the character started meaning "following; inferior (to)", the component 二 in this second translation adds part of the meaning if interpreted as a number.
In all cases, the Kangxi radical "Ice" is a false friend of either "Number two" or two drops representing saliva; however, in some fonts, it is still possible to see 二. An old variant of the character is 𣄭.
Probably from *s- + 二 (OC *njis, “two”), from Proto-Sino-Tibetan *g/s-ni-s (Pulleyblank, 1962; Sagart and Baxter, 2012; STEDT).


二次方程

ㄦˋ ㄘˋ ㄈㄤ ㄔㄥˊ

/ˀɤɻ⁵¹⁻⁵³ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jiː²² t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/ˀɤɻ⁵¹⁻⁵³ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jiː²² t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

(algebra, Mainland China, Hong Kong) quadratic equation


聯立方程

ㄌㄧㄢˊ ㄌㄧˋ ㄈㄤ ㄔㄥˊ

ляньли фанчэн

/li̯ɛn³⁵ li⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/lyːn²¹ lɐp̚² fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/lyːn²¹ laːp̚² fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/li̯ɛn³⁵ li⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/lyːn²¹ lɐp̚² fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/lyːn²¹ laːp̚² fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

simultaneous equations
simultaneous equations


配方

/pʰeɪ̯⁵¹ fɑŋ⁵⁵/

/pʰeɪ̯⁵¹ fɑ̃ɻ⁵⁵/

/pʰuːi̯³³ fɔːŋ⁵⁵/

/pʰᵘɔi³³ fɔŋ³³⁻³³⁵/

/pʰi⁵⁵ foŋ²⁴/

/pʰue²¹⁻⁴¹ hŋ̍⁴⁴/

/pʰue²¹⁻⁵³ hŋ̍⁴⁴/

/pʰue⁴¹⁻⁵⁵⁴ hŋ̍³³/

/pʰe¹¹⁻⁵³ hŋ̍⁴⁴/

/pʰue⁴¹⁻⁵⁵⁴ hɔŋ³³/

/pʰue²¹⁻⁵³ hɔŋ⁴⁴/

/pʰeɪ̯⁵¹ fɑŋ⁵⁵/

/pʰeɪ̯⁵¹ fɑ̃ɻ⁵⁵/

/pʰuːi̯³³ fɔːŋ⁵⁵/

/pʰᵘɔi³³ fɔŋ³³⁻³³⁵/

/pʰi⁵⁵ foŋ²⁴/

/pʰue²¹⁻⁴¹ hŋ̍⁴⁴/

/pʰue²¹⁻⁵³ hŋ̍⁴⁴/

/pʰue⁴¹⁻⁵⁵⁴ hŋ̍³³/

/pʰe¹¹⁻⁵³ hŋ̍⁴⁴/

/pʰue⁴¹⁻⁵⁵⁴ hɔŋ³³/

/pʰue²¹⁻⁵³ hɔŋ⁴⁴/


常數

/ʈ͡ʂʰɑŋ³⁵ ʂu⁵¹/

/sœːŋ²¹ sou̯³³/

/sĩõ⁵⁵⁻¹¹ siau²¹³/

/ʈ͡ʂʰɑŋ³⁵ ʂu⁵¹/

/sœːŋ²¹ sou̯³³/

/sĩõ⁵⁵⁻¹¹ siau²¹³/


展開

/ʈ͡ʂän²¹⁴⁻²¹ kʰaɪ̯⁵⁵/

/t͡siːn³⁵ hɔːi̯⁵⁵/

/tiɛn⁵⁵⁴⁻²⁴ kʰai³³/

/tiɛn⁵³⁻⁴⁴ kʰai⁴⁴/

/tiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵⁵⁴⁻²⁴ kʰui³³/

/tʰiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tʰiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiaŋ⁵²⁻³⁵ kʰui³³/

/ʈ͡ʂän²¹⁴⁻²¹ kʰaɪ̯⁵⁵/

/t͡siːn³⁵ hɔːi̯⁵⁵/

/tiɛn⁵⁵⁴⁻²⁴ kʰai³³/

/tiɛn⁵³⁻⁴⁴ kʰai⁴⁴/

/tiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tiɛn⁵⁵⁴⁻²⁴ kʰui³³/

/tʰiɛn⁵³⁻⁴⁴ kʰui⁴⁴/

/tʰiɛn⁴¹⁻⁴⁴ kʰui⁴⁴/

/tiaŋ⁵²⁻³⁵ kʰui³³/

(algebra) to expand (to rewrite (an expression) as a longer, yet equivalent sum of terms)


共軛

/kʊŋ⁵¹⁻⁵³ ˀɤ⁵¹/

/kʊŋ⁵¹⁻⁵³ ˀɤ⁵¹/

Based on the calque of Latin coniugāre and inflected forms, from con- (“共”) + iugum (“軛”).

(algebra, of a complex number) complex conjugate


線性方程

ㄒㄧㄢˋ ㄒㄧㄥˋ ㄈㄤ ㄔㄥˊ

сяньсин фанчэн

/ɕi̯ɛn⁵¹⁻⁵³ ɕiŋ⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/siːn³³ sɪŋ³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/ɕi̯ɛn⁵¹⁻⁵³ ɕiŋ⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/siːn³³ sɪŋ³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/


一次方程

/i⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jɐt̚⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/i⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/jɐt̚⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/


三次方程

ㄙㄢ ㄘˋ ㄈㄤ ㄔㄥˊ

саньцы фанчэн

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/saːm⁵⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵/

/saːm⁵⁵ t͡sʰiː³³ fɔːŋ⁵⁵ t͡sʰɪŋ²¹/

(algebra, Mainland China, Hong Kong) cubic equation


判別式

/pʰän⁵¹ pi̯ɛ³⁵ ʂʐ̩⁵¹/

/pʰuːn³³ piːt̚² sɪk̚⁵/

/pʰän⁵¹ pi̯ɛ³⁵ ʂʐ̩⁵¹/

/pʰuːn³³ piːt̚² sɪk̚⁵/

(algebra) discriminant (function of a polynomial's coefficients)


方程組

/fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ t͡su²¹⁴⁻²¹⁽⁴⁾/

/fɔːŋ⁵⁵ t͡sʰɪŋ²¹ t͡sou̯³⁵/

/fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ t͡su²¹⁴⁻²¹⁽⁴⁾/

/fɔːŋ⁵⁵ t͡sʰɪŋ²¹ t͡sou̯³⁵/


聯立

(algebra) to list (two or more equations) together and solve simultaneously


閉包

/pi⁵¹ pɑʊ̯⁵⁵/

/pi⁵¹ pɑʊ̯⁵⁵/

(programming, algebra, topology) closure


代數結構

ㄉㄞˋ ㄕㄨˋ ㄐㄧㄝˊ ㄍㄡˋ

дайшу цзегоу

/taɪ̯⁵¹⁻⁵³ ʂu⁵¹ t͡ɕi̯ɛ³⁵ koʊ̯⁵¹/

/tɔːi̯²² sou̯³³ kiːt̚³ kʰɐu̯³³/

/tɔːi̯²² sou̯³³ kiːt̚³ kɐu̯³³/

/taɪ̯⁵¹⁻⁵³ ʂu⁵¹ t͡ɕi̯ɛ³⁵ koʊ̯⁵¹/

/tɔːi̯²² sou̯³³ kiːt̚³ kʰɐu̯³³/

/tɔːi̯²² sou̯³³ kiːt̚³ kɐu̯³³/


進位

/t͡ɕin⁵¹⁻⁵³ weɪ̯⁵¹/

/t͡sɵn³³ wɐi̯²²⁻³⁵/

/t͡sɵn³³ wɐi̯²²/

/t͡ɕin⁵¹⁻⁵³ weɪ̯⁵¹/

/t͡sɵn³³ wɐi̯²²⁻³⁵/

/t͡sɵn³³ wɐi̯²²/

(algebra) to carry (number during addition, etc.); to carry over


交換環

ㄐㄧㄠ ㄏㄨㄢˋ ㄏㄨㄢˊ

цзяохуаньхуань

/t͡ɕi̯ɑʊ̯⁵⁵ xu̯än⁵¹ xu̯än³⁵/

/t͡ɕi̯ɑʊ̯⁵⁵ xu̯än⁵¹ xu̯än³⁵/


整環

/ʈ͡ʂɤŋ²¹⁴⁻²¹ xu̯än³⁵/

/t͡sɪŋ³⁵ waːn²¹/

/ʈ͡ʂɤŋ²¹⁴⁻²¹ xu̯än³⁵/

/t͡sɪŋ³⁵ waːn²¹/


阿貝爾群

ㄚ ㄅㄟˋ ㄦˇ ㄑㄩㄣˊ

/ˀä⁵⁵ peɪ̯⁵¹ ˀɤɻ²¹⁴⁻²¹ t͡ɕʰyn³⁵/

/ˀä⁵⁵ peɪ̯⁵¹ ˀɤɻ²¹⁴⁻²¹ t͡ɕʰyn³⁵/


陪集

/pʰeɪ̯³⁵ t͡ɕi³⁵/

/pʰeɪ̯³⁵ t͡ɕi³⁵/


二次方程式

ㄦˋ ㄘˋ ㄈㄤ ㄔㄥˊ ㄕˋ

эрцы фанчэнши

/ˀɤɻ⁵¹⁻⁵³ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ ʂʐ̩⁵¹/


超越次數

ㄔㄠ ㄩㄝˋ ㄘˋ ㄕㄨˋ

/ʈ͡ʂʰɑʊ̯⁵⁵ ɥɛ⁵¹⁻⁵³ t͡sʰz̩⁵¹⁻⁵³ ʂu⁵¹/

/ʈ͡ʂʰɑʊ̯⁵⁵ ɥɛ⁵¹⁻⁵³ t͡sʰz̩⁵¹⁻⁵³ ʂu⁵¹/

Particularly, the extension field of K is algebraic if and only if its transcendence degree on K is zero.

2003, 朱尧辰 (Zhu Yaochen), 徐广善 (Xu Guangshan), 《超越数引论》 [Introduction to Transcendental Number]

Tèbié, K de kuòyù shì dàishù de, dāngqiějǐndāng tā zài K shàng de chāoyuè cìshù wèi líng.

特別，K的擴域是代數的，當且僅當它在K上的超越次數為零。

特别，K的扩域是代数的，当且仅当它在K上的超越次数为零。

transcendence degree
transcendence degree

(algebra, field theory, of a field extension) transcendence degree


增根

/t͡sɤŋ⁵⁵ kən⁵⁵/

/t͡sɤŋ⁵⁵ kən⁵⁵/


移項

/i³⁵ ɕi̯ɑŋ⁵¹/

/jiː²¹ hɔːŋ²²/

/i³⁵ ɕi̯ɑŋ⁵¹/

/jiː²¹ hɔːŋ²²/

(intransitive, algebra) to transpose terms


布爾函數

ㄅㄨˋ ㄦˇ ㄏㄢˊ ㄕㄨˋ

/pu⁵¹ ˀɤɻ²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

/pu⁵¹ ˀɤɻ²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

(algebra, logic, computing) Boolean function


餘式定理

ㄩˊ ㄕˋ ㄉㄧㄥˋ ㄌㄧˇ

/y³⁵ ʂʐ̩⁵¹⁻⁵³ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/jyː²¹ sɪk̚⁵ tɪŋ²² lei̯¹³/

/y³⁵ ʂʐ̩⁵¹⁻⁵³ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/jyː²¹ sɪk̚⁵ tɪŋ²² lei̯¹³/

(algebra) polynomial remainder theorem; remainder theorem


因式定理

ㄧㄣ ㄕˋ ㄉㄧㄥˋ ㄌㄧˇ

/in⁵⁵ ʂʐ̩⁵¹⁻⁵³ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/jɐn⁵⁵ sɪk̚⁵ tɪŋ²² lei̯¹³/

/in⁵⁵ ʂʐ̩⁵¹⁻⁵³ tiŋ⁵¹ li²¹⁴⁻²¹⁽⁴⁾/

/jɐn⁵⁵ sɪk̚⁵ tɪŋ²² lei̯¹³/


數域

/ʂu⁵¹⁻⁵³ y⁵¹/

/ʂu⁵¹⁻⁵³ y⁵¹/


外代數

/waɪ̯⁵¹⁻⁵³ taɪ̯⁵¹⁻⁵³ ʂu⁵¹/

/waɪ̯⁵¹⁻⁵³ taɪ̯⁵¹⁻⁵³ ʂu⁵¹/


三次方程式

ㄙㄢ ㄘˋ ㄈㄤ ㄔㄥˊ ㄕˋ

саньцы фанчэнши

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ ʂʐ̩⁵¹/

/sän⁵⁵ t͡sʰz̩⁵¹ fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ ʂʐ̩⁵¹/